Provably Robust Simplification of Component Trees of Multidimensional Images

Abstract

We are interested in translating n-dimensional arrays of real numbers (images) into simpler structures that nevertheless capture the topological/geometrical essence of the objects in the images. In the case n=3 these structures may be used as descriptors of images in macromolecular databases. A foreground component tree structure (FCTS) contains all the information on the relationships between connected components when the image is thresholded at various levels. But unsimplified FCTSs are too sensitive to errors in the image to be good descriptors. This chapter presents a method of simplifying FCTSs which can be proved to be robust in the sense of producing essentially the same simplifications in the presence of small perturbations. We demonstrate the potential applicability of our methodology to macromolecular databases by showing that the simplified FCTSs can be used to distinguish between two slightly different versions of an adenovirus.

Appendices

Appendix A: Some Properties of Simplification Steps 2 and 3, and a Proof of the Correctness of Algorithm 1

1.1 A.1 Properties of Simplification Step 2

Here we prove the main result of Sect. 2.4.2, and establish other properties of simplification step 2 that are used in our proof of the Main Theorem.

Lemma A1

Let be any κ-FCTS, let λ>0, and let s and s′ be any two distinct leaves of a κ-FCTS that results from pruning $\boldsymbol{\mathfrak{F}}_{\mathrm{in}}$ by removing branches of length ≤λ. Then (regardless of which ℓ _in-increasing enumeration of is used to perform the pruning):

• (i)
• (ii)

Proof

The hypotheses imply that properties P1–P4 hold with respect to some ℓ _in-increasing enumeration of . It follows from P4 that, for all , every node in is also a node in . Therefore is the same set regardless of whether or . So is the same node regardless of whether or , since is just the element of that is a descendant in of every element of that set. Hence (i) holds.

To prove (ii), we may assume without loss of generality that, in the ℓ _in-increasing leaf enumeration that is used for pruning, s occurs later than s′. (This assumption implies that min(ℓ _in(s),ℓ _in(s′))=ℓ _in(s′).) Then, since , property P3 implies that , which is equivalent to:

(A1)

But (A1) is equivalent to assertion (ii), because of assertion (i) and the fact that ℓ _out is just the restriction of ℓ _in to . □

Corollary A2

Let λ be any positive value, and $\boldsymbol{\mathfrak{F}}_{\mathrm{out}}$ any κ-FCTS that results from pruning a κ-FCTS $\boldsymbol {\mathfrak{F}}_{\mathrm{in}}$ by removing branches of length ≤λ. Then, for all $\mathbf{v}\in\mathbf{Crit}(\boldsymbol{\mathfrak {F}}_{\mathrm{out}}) \setminus\mathbf{Leaves} (\boldsymbol{\mathfrak{F}}_{\mathrm{out}})$, we have that $\mathbf{v}\in\mathbf{Crit}(\boldsymbol{\mathfrak {F}}_{\mathrm{in}}) \setminus\mathbf{Leaves}(\boldsymbol{\mathfrak{F}}_{\mathrm{in}})$ and $\mathrm{depth}_{\boldsymbol{\mathfrak{F}}_{\mathrm{out}}}(\mathbf {v}) > \lambda$.

Proof

Let , and let $\mathbf{v}\in\mathbf{Crit}(\boldsymbol{\mathfrak{F}}_{\mathrm {out}}) \setminus\mathbf{Leaves}(\boldsymbol{\mathfrak{F}}_{\mathrm {out}})$. Then for some distinct leaves s and s′ of $\boldsymbol{\mathfrak {F}}_{\mathrm{out}}$. Now (by assertion (i) of Lemma A1), and so $\mathbf{v}\in \mathbf{Crit}(\boldsymbol{\mathfrak{F}}_{\mathrm{in}}) \setminus\mathbf{Leaves}(\boldsymbol{\mathfrak {F}}_{\mathrm{in}})$. Moreover, we have that , where the second inequality follows from assertion (ii) of Lemma A1. □

Lemma A3

Let be a κ-FCTS, let λ>0, and let be the κ-FCTS that results from pruning $\boldsymbol{\mathfrak{F}}_{\mathrm{in}}$ by removing branches of length ≤λ using an ℓ _in-increasing leaf enumeration σ=(leaf[1],…,leaf[n]) of . Then:

• (a) For all , .

• (b) For all , if and only if .

• (c) For all , $\mathrm{depth}_{\boldsymbol{\mathfrak{F}}_{\mathrm{out} }}(\mathbf{v}) = \mathrm{depth}_{\boldsymbol{\mathfrak{F}}_{\mathrm {in}}}(\mathbf{v})$.

Proof

For brevity, we write lastLeaf _σ (v) for . Evidently, (a) follows from P4, and the “if” part of (b) follows from (a). To establish the “only if” part of (b), let , and let leaf[i]=lastLeaf _σ (v). We need to show that . If i=n then this is true (by property P2), so let us assume i<n. Let j be any element of the set {i+1,…,n} (so that ). Now we claim that leaf[j] must satisfy .

To see this, let leaf[k] be any leaf of ; such a leaf must exist, by P4. As leaf[i]=lastLeaf _σ (v), we have that i≥k and ℓ _in(leaf[i])≥ℓ _in(leaf[k]). As j∈{i+1,…,n}, we have that j∈{k+1,…,n}. Therefore, since , property P3 implies that:

(A2)

But, since leaf[i] and leaf[k] are leaves of but leaf[j] is not,

and (since ℓ _in(leaf[i])≥ℓ _in(leaf[k])) this implies:

This and (A2) imply that our claim is valid (for any j in {i+1,…,n}). The “only if” part of (b) follows from this and property P3.

To prove (c), let . Then (by (b)), and every satisfies ℓ _out(w)=ℓ _in(w)≤ℓ _in(lastLeaf _σ (v))=ℓ _out(lastLeaf _σ (v)).

It follows that $\mathrm{depth}_{\boldsymbol{\mathfrak{F}}_{\mathrm{out}}}(\mathbf {v}) = \ell_{\mathrm{out}}(\mathsf {lastLeaf}_{\sigma} (\mathbf{v})) - \ell_{\mathrm{out}}(\mathbf{v}) = \ell_{\mathrm{in}}(\mathsf {lastLeaf}_{\sigma}(\mathbf{v})) - \ell_{\mathrm{in} }(\mathbf{v}) = \mathrm{depth}_{\boldsymbol{\mathfrak{F}}_{\mathrm{in} }}(\mathbf{v})$. □

Lemma A4

Let be a κ-FCTS, let λ>0, and let be the κ-FCTS that results from pruning $\boldsymbol{\mathfrak{F}}_{\mathrm{in}}$ by removing branches of length ≤λ using an ℓ _in-increasing leaf enumeration σ=(leaf[1],…,leaf[n]) of . Then:

• (a)

• (b) For all $\mathbf{v}\in\mathbf{V}^{\lambda}\langle \boldsymbol{\mathfrak{F}}_{\mathrm{in}}\rangle \setminus\mathbf{V}_{1}^{\lambda}\langle\boldsymbol{\mathfrak {F}}_{\mathrm{in}}\rangle$, .

• (c) For all $\mathbf{v}\in\mathbf{V}_{1}^{\lambda}\langle \boldsymbol{\mathfrak{F}}_{\mathrm{in}}\rangle$, .

Proof

For brevity, we shall write U ^λ, V ^λ, $\mathbf {V}_{1}^{\lambda}$, lastLeaf _σ (v), and Path _σ (v) for $\mathbf{U}^{\lambda}\langle\boldsymbol{\mathfrak{F}}_{\mathrm {in}}\rangle$, $\mathbf{V}^{\lambda}\langle\boldsymbol{\mathfrak{F}}_{\mathrm {in}}\rangle$, $\mathbf{V}_{1}^{\lambda}\langle\boldsymbol{\mathfrak{F}}_{\mathrm {in}}\rangle$, , and .

First, we prove (a). The inclusion follows from Corollary A2 and Lemma A3(c). Moreover, since P4 implies that , we have that if u∈U ^λ. So the other inclusion of (a) will follow if we can show that whenever u∈U ^λ.

Let u be any element of U ^λ, and let leaf[i]=lastLeaf _σ (u). If i=n, then (by property P2) and so (because of P4), as required. Now suppose i<n. Let j be any element of the set {i+1,…,n} (so ). Since leaf[i] is a leaf of but leaf[j] is not, we have that . Hence:

We see from this and property P3 that , and hence (in view of P4) that , as required. This proves (a).

Next, we prove (b). Let v be any node in $\mathbf{V}^{\lambda}\setminus\mathbf {V}_{1}^{\lambda}$. Then it follows from the definitions of V ^λ and $\mathbf{V}_{1}^{\lambda}$ that .

Let . Then , so we have that:

$$ \ell_{\mathrm{in}}\bigl(\mathsf {lastLeaf}_{\sigma}(\mathbf{p}) \bigr) - \ell_{\mathrm {in}}(\mathbf{p}) = \mathrm{depth}_{\boldsymbol{\mathfrak{F}}_{\mathrm{in} }}( \mathbf{p}) > \lambda $$

(A3)

Now $\ell_{\mathrm{in}}(\mathbf{d}) - \ell_{\mathrm{in}}(\mathbf {v}) \leq\mathrm{depth}_{\boldsymbol{\mathfrak{F}} _{\mathrm{in}}}(\mathbf{v})$ for all . Therefore:

(A4)

Here the second inequality follows from the definition of $\mathbf {V}_{1}^{\lambda}$ and the facts that and $\mathbf{v}\in \mathbf{V}^{\lambda}\setminus\mathbf{V}_{1}^{\lambda}$. It follows from (A3) and (A4) that lastLeaf _σ (p) is not a descendant of v in , and so

(A5)

Since , we deduce from (A4) and (A5) that

(A6)

Since and lastLeaf _σ (p)≠lastLeaf _σ (v) (e.g., by (A5)), the leaf lastLeaf _σ (p) must occur later in the ℓ _in-increasing enumeration σ than the leaf lastLeaf _σ (v). This, (A6), and P3 imply that . It now follows from assertion (b) of Lemma A3 that . This and assertion (a) of Lemma A3 imply , which proves (b).

Finally, we prove (c). Let v be any node in $\mathbf{V}_{1}^{\lambda}$. We first make the claim that lastLeaf _σ (v) is a leaf of .

If then the claim is certainly true (by property P2), so let us assume . Let , and let s be any leaf of that occurs later in the ℓ _in-increasing enumeration σ than lastLeaf _σ (v). Then , and so , which implies that:

(A7)

But, since $\mathrm{depth}_{\boldsymbol{\mathfrak{F}}_{\mathrm {in}}}(\mathbf{v}) = \ell_{\mathrm{in} }(\mathsf {lastLeaf}_{\sigma}(\mathbf{v})) - \ell_{\mathrm{in}}(\mathbf{v})$, we also have that

$$ \ell_{\mathrm{in}}\bigl(\mathsf {lastLeaf}_{\sigma}(\mathbf{v}) \bigr) - \ell_{\mathrm {in}}(\mathbf{p}) = \mathrm{depth}_{\boldsymbol{\mathfrak{F}}_{\mathrm{in} }}( \mathbf{v}) + \ell_{\mathrm{in}}(\mathbf{v}) - \ell_{\mathrm {in}}(\mathbf{p}) > \lambda $$

(A8)

where the inequality follows from the definition of $\mathbf {V}_{1}^{\lambda}$ and the facts that and $\mathbf{v}\in \mathbf{V}_{1}^{\lambda}$. Now it follows from (A7) and (A8) that:

Since this is true for every leaf s of that occurs later in the ℓ _in-increasing enumeration σ than lastLeaf _σ (v), our claim is justified (by property P3).

If w is any node in Path _σ (v), then and so it follows from our claim (and P4) that . Thus every node in Path _σ (v) lies in .

It remains only to prove that . To do this, we suppose there is a node and deduce a contradiction. As , we have that and so lastLeaf _σ (v)≠lastLeaf _σ (x). Moreover, each of the nodes lastLeaf _σ (x) and lastLeaf _σ (v) is a leaf of (by Lemma A3(b) and our claim).

Let . Then we have that , , and (by assertion (i) of Lemma A1). The latter implies (as and ); and implies $\mathrm{depth}_{\boldsymbol{\mathfrak{F}}_{\mathrm{in}}} \mathbf {c}\leq\mathrm{depth}_{\boldsymbol{\mathfrak{F}}_{\mathrm{in}}} \mathbf {v}\leq\lambda$ (where the second inequality follows from the fact that $\mathbf{v}\in\mathbf{V}_{1}^{\lambda}\subseteq \mathbf {V}^{\lambda}$). Hence c∉U ^λ. But this contradicts assertion (a) (since ). It follows that x cannot exist, and so our proof of (c) is complete. □

We can now prove the main result of Sect. 2.4.2:

Proposition

Let be a κ-FCTS, let λ>0, and let be the κ-FCTS that results from pruning $\boldsymbol{\mathfrak{F}}_{\mathrm{in}}$ by removing branches of length ≤λ using an ℓ _in-increasing enumeration σ of . Then the nodes of $\boldsymbol{\mathfrak{F}}_{\mathrm{out}}$ consist just of:

• (i) The nodes of $\mathbf{U}^{\lambda}\langle \boldsymbol{\mathfrak{F}}_{\mathrm{in} }\rangle$.

• (ii) The nodes of for each node v in $\mathbf{V}_{1}^{\lambda}\langle\boldsymbol {\mathfrak{F}}_{\mathrm{in} }\rangle$.

Proof

As by Lemma A4(a), on putting and $\boldsymbol{\mathfrak {F}}= \boldsymbol{\mathfrak{F}}_{\mathrm{in}}$ in (2.1) and taking the intersection of each side with we see that:

The proposition follows from this and assertions (b) and (c) of Lemma A4. □

1.2 A.2 Properties of Simplification Step 3

Here we establish some properties of simplification step 3 that are used in our proof of the Main Theorem and our justification of Algorithm 1.

For all $j \in\{1, \dots, |D(\boldsymbol{\mathfrak{F}})|\}$, we see from E1–E5 that $\mathbf{Nodes}(\boldsymbol{\mathfrak{F}}^{\mathbf{crit}}{\langle \delta\rangle}) \subseteq\mathbf{Nodes}(\boldsymbol{\mathfrak{F}}^{\mathbf{crit}}{\langle \delta' \rangle})$ whenever δ≥δ′. It follows that $\boldsymbol{\mathfrak{F}}^{\mathbf{crit}}{\langle\cdot\rangle}$ has the following monotonicity property:

$$ \boldsymbol{\mathfrak{F}}^\mathbf{crit} {\langle\delta \rangle} \sqsubseteq\boldsymbol{\mathfrak{F}}^\mathbf {crit} {\bigl\langle \delta' \bigr\rangle} \quad \mbox{whenever } \delta\geq \delta' $$

(A9)

In addition, $\boldsymbol{\mathfrak{F}}^{\mathbf{crit}}{\langle\cdot \rangle}$ has the following four properties for every λ>0 (as we will explain below):

• E6: For every $\mathbf{c}\in\mathbf{Nodes}(\boldsymbol {\mathfrak{F}}^{\mathbf{crit}}) \setminus(\mathbf{Leaves}(\boldsymbol{\mathfrak{F}}) \cup \{\mathbf{LCN}(\boldsymbol{\mathfrak{F}})\} \cup\{\mathbf {root}(\boldsymbol{\mathfrak{F}})\})$ and every $i \in\{0, \dots, |D(\boldsymbol{\mathfrak{F}})|-1\}$, $\mathbf{c}\in\mathbf{Nodes}(\boldsymbol{\mathfrak{F}}^{\mathbf{crit}}{\langle d^{\boldsymbol{\mathfrak{F}}}_{i+1} \rangle})$ if and only if, for every j∈{0,…,i}, $\ell(\mathbf{c}) - \ell(\mathbf{parent}_{\boldsymbol{\mathfrak {F}}^{\mathbf{crit}}{\langle d^{\boldsymbol{\mathfrak{F}} }_{j} \rangle}}(\mathbf{c})) > d^{\boldsymbol{\mathfrak{F}}}_{j+1}$.

• E7: For every $\mathbf{c}\in\mathbf{Nodes}(\boldsymbol {\mathfrak{F}}^{\mathbf{crit}}) \setminus(\mathbf{Leaves}(\boldsymbol{\mathfrak{F}}) \cup \{\mathbf{LCN}(\boldsymbol{\mathfrak{F}})\} \cup\{\mathbf {root}(\boldsymbol{\mathfrak{F}})\})$, $\mathbf{c}\in\mathbf{Nodes}(\boldsymbol{\mathfrak{F}}^{\mathbf{crit}}{\langle\lambda\rangle})$ if and only if there is no critical proper ancestor c′ of c in $\boldsymbol{\mathfrak{F}}$ such that ℓ(c)−ℓ(c′)≤λ and $\mathbf{c}' \in\mathbf{Nodes}(\boldsymbol{\mathfrak{F}}^{\mathbf{crit}}{\langle\operatorname{pred}_{\boldsymbol{\mathfrak{F}}}(\ell (\mathbf{c}) - \ell(\mathbf{c}')) \rangle})$.

• E8: For every $\mathbf{c}\in\mathbf{Nodes}(\boldsymbol {\mathfrak{F}}^{\mathbf{crit}}) \setminus(\mathbf{Leaves}(\boldsymbol{\mathfrak{F}}) \cup \{\mathbf{LCN}(\boldsymbol{\mathfrak{F}})\} \cup\{\mathbf {root}(\boldsymbol{\mathfrak{F}})\})$, $\mathbf{c}\in\mathbf{Nodes}(\boldsymbol{\mathfrak{F}}^{\mathbf{crit}}{\langle\lambda\rangle})$ if $\ell(\mathbf{c}) - \ell(\mathbf{parent}_{\boldsymbol{\mathfrak {F}}^{\mathbf{crit}}}(\mathbf {c})) > \lambda$.

• E9: For every $\mathbf{c}\in\mathbf{Nodes}(\boldsymbol {\mathfrak{F}}^{\mathbf{crit}}) \setminus(\mathbf{Leaves}(\boldsymbol{\mathfrak{F}}) \cup \{\mathbf{LCN}(\boldsymbol{\mathfrak{F}})\} \cup\{\mathbf {root}(\boldsymbol{\mathfrak{F}})\})$, if $\mathbf{c}\in\mathbf{Nodes}(\boldsymbol{\mathfrak{F}}^{\mathbf{crit}}{\langle\lambda\rangle})$ then $\ell(\mathbf{c}) - \ell(\mathbf{parent}_{\boldsymbol{\mathfrak {F}}^{\mathbf{crit}}{\langle \lambda\rangle}}(\mathbf{c})) > \lambda$.

Our proof of the correctness of Algorithm 1 will be based on property E7. However, E1–E3, E8, and E9 are the only properties of simplification step 3 that will be used in our proof of the Main Theorem.

E6 is easily deduced from E5 by induction on i. Now we establish E7–E9. Let $\mathbf{c}\in\mathbf{Nodes}(\boldsymbol{\mathfrak{F}}^{\mathbf{crit}}) \setminus(\mathbf{Leaves}(\boldsymbol{\mathfrak{F}}) \cup\{\mathbf{LCN}(\boldsymbol{\mathfrak{F}})\} \cup\{\mathbf{root}(\boldsymbol{\mathfrak{F}})\})$, and let λ be any positive value. We first claim that, for any critical proper ancestor c′ of c in $\boldsymbol{\mathfrak{F}}$, the following four conditions are equivalent:

• (a) There is some $j \in\{0, \dots, |D(\boldsymbol{\mathfrak {F}})|-1\}$ such that $\ell(\mathbf{c}) - \ell(\mathbf{c}') \leq d^{\boldsymbol {\mathfrak{F}}}_{j+1} \leq \lambda$ and $\mathbf{c}' \in \mathbf{Nodes}(\boldsymbol{\mathfrak{F}}^{\mathbf{crit}}{\langle d^{\boldsymbol{\mathfrak{F}}}_{j} \rangle})$.

• (b) There is some $j \in\{0, \dots, |D(\boldsymbol{\mathfrak {F}})|-1\}$ such that $\ell(\mathbf{c}) - \ell(\mathbf{c}') \leq d^{\boldsymbol {\mathfrak{F}}}_{j+1} \leq \lambda$ and $\mathbf{c}' \in\mathbf{Nodes}(\boldsymbol{\mathfrak {F}}^{\mathbf{crit}}{\langle\operatorname{pred}_{\boldsymbol {\mathfrak{F}} }(\ell(\mathbf{c}) - \ell(\mathbf{c}')) \rangle})$.

• (c) ℓ(c)−ℓ(c′)≤λ and $\mathbf{c}' \in\mathbf{Nodes}(\boldsymbol{\mathfrak{F}}^{\mathbf{crit}}{\langle\operatorname{pred}_{\boldsymbol{\mathfrak {F}}}(\ell(\mathbf{c}) - \ell(\mathbf {c}')) \rangle})$.

• (d) There is some $j \in\{0, \dots, |D(\boldsymbol{\mathfrak {F}})|-1\}$ such that $\ell(\mathbf{c}) - \ell(\mathbf{c}') = d^{\boldsymbol{\mathfrak {F}}}_{j+1} \leq\lambda$ and $\mathbf{c}' \in\mathbf{Nodes} (\boldsymbol{\mathfrak{F}}^{\mathbf{crit}}{\langle d^{\boldsymbol {\mathfrak{F}}}_{j} \rangle})$.

Here (a) implies (b) because of the monotonicity property (A9) and the fact that if $\ell(\mathbf{c}) - \ell(\mathbf{c}') \leq d^{\boldsymbol{\mathfrak{F}} }_{j+1}$ then $\operatorname{pred}_{\boldsymbol{\mathfrak{F}}}(\ell(\mathbf{c}) - \ell(\mathbf{c}')) \leq d^{\boldsymbol{\mathfrak{F}}}_{j}$. Evidently, (b) implies (c), and (d) implies (a). For any critical proper ancestor c′ of c in $\boldsymbol{\mathfrak{F}}$, $\ell(\mathbf{c}) - \ell(\mathbf{c}') = d^{\boldsymbol{\mathfrak {F}}}_{j+1}$ and $\operatorname{pred}_{\boldsymbol{\mathfrak{F}}}(\ell(\mathbf{c}) - \ell(\mathbf{c}')) = d^{\boldsymbol{\mathfrak{F}}}_{j}$ for some $j \in\{0, \dots, |D(\boldsymbol{\mathfrak{F}})|-1\}$, and so (c) implies (d). This justifies our claim that (a)–(d) are equivalent.

Next, we observe that $\mathbf{c}\in\mathbf{Nodes}(\boldsymbol{\mathfrak{F}}^{\mathbf{crit}}{\langle\lambda\rangle})$ holds if and only if c satisfies $\ell(\mathbf{c}) - \ell(\mathbf{parent}_{\boldsymbol{\mathfrak {F}}^{\mathbf{crit}}{\langle d^{\boldsymbol{\mathfrak{F}} }_{j} \rangle}}(\mathbf{c})) > d^{\boldsymbol{\mathfrak{F}}}_{j+1}$ for all $j \in\{0, \dots, |D(\boldsymbol{\mathfrak{F}})|-1\}$ such that $d^{\boldsymbol{\mathfrak{F}}}_{j+1} \leq \lambda$. (This follows from E6 when $\lambda\in D(\boldsymbol{\mathfrak {F}})$. It remains true if $\lambda\notin D(\boldsymbol{\mathfrak{F}})$, because of E4.) So $\mathbf{c}\notin\mathbf{Nodes}(\boldsymbol{\mathfrak {F}}^{\mathbf{crit}}{\langle\lambda\rangle})$ just if there is some $j \in\{0, \dots, |D(\boldsymbol{\mathfrak{F}})|-1\}$ such that $\ell(\mathbf{c}) - \ell(\mathbf{parent}_{\boldsymbol{\mathfrak {F}}^{\mathbf{crit}}{\langle d^{\boldsymbol{\mathfrak{F}} }_{j} \rangle}}(\mathbf{c})) \leq d^{\boldsymbol{\mathfrak{F}}}_{j+1} \leq\lambda$. Thus $\mathbf{c}\notin\mathbf{Nodes}(\boldsymbol{\mathfrak {F}}^{\mathbf{crit}}{\langle\lambda \rangle})$ just if (a) holds for some critical proper ancestor c′ of c in $\boldsymbol{\mathfrak{F}}$. Equivalently, $\mathbf{c}\notin\mathbf{Nodes}(\boldsymbol{\mathfrak {F}}^{\mathbf{crit}}{\langle \lambda\rangle})$ just if (c) holds for some critical proper ancestor c′ of c in $\boldsymbol{\mathfrak{F}}$. This proves E7. E8 follows from the “if” part of E7.

Suppose the node c violated E9. Then $\mathbf{c}\in\mathbf{Nodes}(\boldsymbol{\mathfrak{F}}^{\mathbf{crit}}{\langle\lambda\rangle})$. Moreover, when $\mathbf{c}' = \mathbf{parent}_{\boldsymbol{\mathfrak {F}}^{\mathbf{crit}}{\langle\lambda \rangle}}(\mathbf{c})$ we would have that ℓ(c)−ℓ(c′)≤λ and also that $\mathbf{c}' \in\mathbf{Nodes}(\boldsymbol{\mathfrak{F}}^{\mathbf{crit}}{\langle\operatorname{pred}_{\boldsymbol{\mathfrak{F}}}(\ell (\mathbf{c}) - \ell(\mathbf{c}')) \rangle})$, where the latter follows from the former, the fact that $\mathbf{c}' \in\mathbf{Nodes}(\boldsymbol{\mathfrak{F}}^{\mathbf{crit}}{\langle\lambda\rangle})$, and the monotonicity property (A9). But this would contradict the “only if” part of E7. So E9 holds.

1.3 A.3 Justification of Algorithm 1

The correctness of Algorithm 1 will be deduced from Lemma A5 and Corollary A6 below.

Let be any κ-FCTS, and let c be any node of $\boldsymbol{\mathfrak{F}}^{\mathbf{crit}}$. Then we define $\delta_{\lambda}(\mathbf{c},\boldsymbol{\mathfrak {F}}) = \infty$ if $\mathbf{c}\in\mathbf{Leaves}(\boldsymbol{\mathfrak{F}}) \cup\{ \mathbf{LCN}(\boldsymbol{\mathfrak{F}})\} \cup\{\mathbf {root}(\boldsymbol{\mathfrak{F}})\}$, and we define $\delta_{\lambda}(\mathbf{c},\boldsymbol{\mathfrak {F}}) = \ell(\mathbf{c}) - \ell(\mathbf{a}_{\lambda}(\mathbf{c},\boldsymbol{\mathfrak{F}}))$ otherwise, where $\mathbf{a}_{\lambda}(\mathbf{c},\boldsymbol {\mathfrak{F}})$ is the closest critical proper ancestor c′ of c in $\boldsymbol{\mathfrak{F}}$ such that

$\mathbf{a}_{\lambda}(\mathbf{c},\boldsymbol{\mathfrak{F}})$ exists for all $\mathbf{c}\in\mathbf{Nodes}(\boldsymbol{\mathfrak{F}}^{\mathbf{crit}}) \setminus (\mathbf{Leaves}(\boldsymbol{\mathfrak{F}}) \cup\{\mathbf {LCN}(\boldsymbol{\mathfrak{F}})\} \cup\{\mathbf{root}(\boldsymbol {\mathfrak{F}})\})$, because when $\mathbf{c}' = \mathbf{LCN}(\boldsymbol{\mathfrak {F}})$ we see from E2 that $\mathbf {c}' \in\mathbf{Nodes}(\boldsymbol{\mathfrak{F}}^{\mathbf{crit}}{\langle\mu\rangle})$ for every μ≥0 and so c′ must satisfy the “either” or the “or” condition. Now $\delta_{\lambda}(\cdot,\boldsymbol{\mathfrak{F}})$ satisfies the following condition:

Lemma A5

Let 0≤μ≤λ and let be any κ-FCTS. Then for all $\mathbf{c}\in\mathbf{Nodes}(\boldsymbol{\mathfrak {F}}^{\mathbf{crit}})$ we have that $\delta_{\lambda}(\mathbf{c},\boldsymbol{\mathfrak{F}}) > \mu$ if and only if $\mathbf{c}\in\mathbf{Nodes}(\boldsymbol{\mathfrak{F}}^{\mathbf{crit}}{\langle\mu\rangle})$.

Proof

Suppose $\mathbf{c}\in\mathbf{Nodes}(\boldsymbol{\mathfrak {F}}^{\mathbf{crit}}) \setminus(\mathbf{Leaves} (\boldsymbol{\mathfrak{F}}) \cup\{\mathbf{LCN}(\boldsymbol {\mathfrak{F}})\} \cup\{\mathbf{root}(\boldsymbol{\mathfrak{F}})\})$. Then $\delta_{\lambda}(\mathbf{c},\boldsymbol{\mathfrak{F}}) > \mu$ holds just if $\ell (\mathbf{c}) - \ell(\mathbf{a}_{\lambda}(\mathbf{c},\boldsymbol{\mathfrak{F}})) > \mu$, and since μ≤λ we see from the definition of $\mathbf{a}_{\lambda}(\mathbf {c},\boldsymbol{\mathfrak{F}})$ that this holds just if no critical proper ancestor c′ of c in $\boldsymbol{\mathfrak{F}}$ satisfies ℓ(c)−ℓ(c′)≤μ and $\mathbf{c}' \in\mathbf{Nodes}(\boldsymbol{\mathfrak{F}}^{\mathbf{crit}}{\langle\operatorname{pred}_{\boldsymbol{\mathfrak{F}}}(\ell (\mathbf{c}) - \ell(\mathbf{c}')) \rangle})$. So in this case the lemma follows from E7.

The lemma also holds if $\mathbf{c}\in\mathbf{Leaves}(\boldsymbol {\mathfrak{F}}) \cup\{\mathbf{LCN}(\boldsymbol{\mathfrak{F}} )\} \cup\{\mathbf{root} (\boldsymbol{\mathfrak{F}})\}$, because in that case $\delta_{\lambda}(\mathbf{c},\boldsymbol{\mathfrak{F}}) = \infty> \mu$ and E1–E3 imply $\mathbf{c}\in\mathbf{Nodes}(\boldsymbol{\mathfrak{F}}^{\mathbf{crit}}{\langle\mu\rangle})$. □

Corollary A6

Let λ be any positive value, let be any κ-FCTS, and let $\mathbf{c}\in\mathbf{Nodes}(\boldsymbol{\mathfrak {F}}^{\mathbf{crit}}) \setminus(\mathbf{Leaves} (\boldsymbol{\mathfrak{F}}) \cup\{\mathbf{LCN}(\boldsymbol {\mathfrak{F}})\} \cup\{\mathbf{root}(\boldsymbol{\mathfrak{F}})\})$. Then $\delta_{\lambda}(\mathbf{c},\boldsymbol{\mathfrak{F}}) = \ell (\mathbf{c}) - \ell(\mathbf{a} )$, where a is the closest critical proper ancestor c′ of c in $\boldsymbol{\mathfrak{F}}$ such that

$$\begin{array}{@{}l@{\quad}l} \mbox{\textbf{either}}&\ell(\mathbf{c}) - \ell\bigl(\mathbf {c}'\bigr) > \lambda\\\noalign{\vspace{3pt}} \mbox{\textbf{or}}&\ell(\mathbf{c}) - \ell\bigl(\mathbf{c}'\bigr) \leq\lambda\mbox{ \textit{and} }\ell(\mathbf{c}) - \ell\bigl(\mathbf{c}'\bigr) \leq\delta_\lambda \bigl(\mathbf{c}',\boldsymbol{\mathfrak{F}}\bigr) \end{array}$$

Proof

We just have to show that $\mathbf{a}= \mathbf{a}_{\lambda}(\mathbf {c},\boldsymbol{\mathfrak{F}})$. The definition of $\mathbf{a}_{\lambda}(\mathbf{c},\boldsymbol {\mathfrak{F}})$ differs from the definition of a only in the or condition “$\ell(\mathbf{c}) - \ell(\mathbf{c}') \leq\lambda \mbox{ and } \mathbf{c}' \in\mathbf{Nodes}(\boldsymbol{\mathfrak {F}}^{\mathbf{crit}}{\langle\operatorname{pred}_{\boldsymbol{\mathfrak{F}}}(\ell(\mathbf{c}) - \ell (\mathbf{c}')) \rangle})$”.

On putting $\mu= \operatorname{pred}_{\boldsymbol{\mathfrak {F}}}(\ell(\mathbf{c}) - \ell(\mathbf {c}'))$ in Lemma A5, we see that this condition holds if and only if ℓ(c)−ℓ(c′)≤λ and $\operatorname{pred}_{\boldsymbol{\mathfrak{F}}}(\ell (\mathbf{c}) - \ell(\mathbf{c}')) < \delta_{\lambda}(\mathbf {c}',\boldsymbol{\mathfrak{F}})$, which is equivalent to the or condition in the definition of a (because either $\delta_{\lambda}(\mathbf{c}',\boldsymbol{\mathfrak {F}}) = \ell(\mathbf {c}) - \ell(\mathbf{a}_{\lambda}(\mathbf{c}',\boldsymbol{\mathfrak{F}})) \in D(\boldsymbol{\mathfrak{F}})$ or $\delta_{\lambda}(\mathbf{c}',\boldsymbol{\mathfrak{F}}) = \infty$). So $\mathbf{a}= \mathbf{a}_{\lambda}(\mathbf{c},\boldsymbol {\mathfrak{F}})$, as required. □

We can now explain why Algorithm 1 is correct. The algorithm sets to a clone of . Writing $\boldsymbol{\mathfrak{F}}$ for , we claim that the label c.label given by the algorithm to each node c of $\boldsymbol{\mathfrak{F}}= \boldsymbol{\mathfrak{F}}^{\mathbf{crit}}$ is just the value $\delta_{\lambda}(\mathbf{c},\boldsymbol{\mathfrak{F}})$. Assuming this claim is valid, the correctness of the algorithm follows from Lemma A5. So it remains only to verify the claim.

The claim is certainly valid if c is $\mathbf{root}(\boldsymbol{\mathfrak{F}})$ or $\mathbf {LCN}(\boldsymbol{\mathfrak{F}})$, because those nodes are given the label ∞.

We see that the algorithm does a top-down traversal of , during which the procedure labelDescendants is executed once for each proper descendant c of $\mathbf {LCN}(\boldsymbol{\mathfrak{F}})$ in $\boldsymbol{\mathfrak{F}}$. When labelDescendants is executed for such a node c that is a leaf, it gives c the label ∞. So the claim is valid for each proper descendant c of $\mathbf{LCN}(\boldsymbol {\mathfrak{F}})$ that is a leaf.

When labelDescendants is executed for a proper descendant c of $\mathbf{LCN}(\boldsymbol {\mathfrak{F}})$ that is not a leaf, the repeat loop in the procedure is executed. It follows from Corollary A6 that this loop labels c with the value $\delta_{\lambda}(\mathbf {c},\boldsymbol{\mathfrak{F}})$. (Note that, when the loop is executed, $\mathbf{c}'.\mathsf {label}= \delta_{\lambda}(\mathbf{c}',\boldsymbol {\mathfrak{F}})$ for each proper ancestor c′ of c in $\boldsymbol {\mathfrak{F}}$.) Therefore the claim is also valid for each proper descendant c of $\mathbf{LCN}(\boldsymbol {\mathfrak{F}})$ that is not a leaf.

Thus the claim is valid for all nodes c of $\boldsymbol {\mathfrak{F}}= \boldsymbol{\mathfrak{F}}^{\mathbf{crit}}$, and Algorithm 1 is correct.

Appendix B: A Constructive Proof of Theorem 1

For any adjacency relation κ, any image I whose domain is finite and κ-connected, any λ>0, and any integer k≥0, let us say that the image I is (λ,k)-good with respect to κ if Λ _κ (I)>λ and K _κ (I)>k. Also, let us say that an image I′ is an ε-perturbation of an image I if I′ has the same domain as I and ∥I′−I∥_∞≤ε. Then Theorem 1 can be deduced from the following lemma:

Fundamental Lemma

Let κ be any adjacency relation and an image whose domain is finite and κ-connected. Let ε be a positive value, let k be a nonnegative integer for which I _good is (4ε,k)-good with respect to κ, and let I′ be an ε-perturbation of I _good . Then there is an essential isomorphism of FCTS _κ (I _good ) to the (2ε,k)-simplification of FCTS _κ (I′) that is level-preserving to within ε.

Proof of Theorem 1, assuming the Fundamental Lemma is valid

Suppose I, λ, and k satisfy the hypotheses of Theorem 1, so that 0<λ<Λ _κ (I)/2 and 0≤k<K _κ (I). Let I′ be any image that satisfies the conditions stated in the theorem (i.e., let I′ be any image whose domain is the same as that of I and which satisfies the condition ∥I′−I∥_∞≤λ/2). Then we need to show that the conclusion of Theorem 1 holds—i.e., that there is an essential isomorphism of the (λ,k)-simplification of FCTS _κ (I′) to FCTS _κ (I) that is level-preserving to within λ/2. We now deduce this from the Fundamental Lemma.

Let I _good =I, and let ε=λ/2. Then 4ε=2λ<Λ _κ (I)=Λ _κ (I _good ) and k<K _κ (I)=K _κ (I _good ), so that I _good is (4ε,k)-good with respect to κ. We also have that ∥I′−I _good ∥_∞=∥I′−I∥_∞≤λ/2=ε, so that I′ is a ε-perturbation of I _good . Thus I _good =I and I′ satisfy the hypotheses of the Fundamental Lemma, and must therefore satisfy the conclusion of the lemma, which implies the conclusion of Theorem 1 since 2ε=λ. □

We now prove the Fundamental Lemma by constructing an explicit essential isomorphism of FCTS _κ (I _good ) to the (2ε,k)-simplification of FCTS _κ (I′) that is level-preserving to within ε.

Let , and let . Let be the κ-FCTS that results from pruning $\boldsymbol{\mathfrak{F}}'$ by removing nodes of size ≤k, and let I ₁ be the image $\mathit{I}_{\boldsymbol {\mathfrak{F}}_{1}}$, so that $\boldsymbol{\mathfrak{F}}_{1} = \mathbf {FCTS}_{{\kappa}} (\mathit{I}_{1})$. Let be the κ-FCTS that results from pruning $\boldsymbol{\mathfrak{F}}_{1}$ by removing branches of length ≤2ε, and let be the κ-FCTS that results from eliminating internal edges of length ≤2ε from $\boldsymbol{\mathfrak{F}}_{2}^{\mathbf{crit}}$. Then is the (2ε,k)-simplification of FCTS _κ (I′), so what we want to do is to construct an essential isomorphism of $\boldsymbol{\mathfrak {F}}_{\mathbf{good}}$ to $\boldsymbol{\mathfrak{F}}_{3}$ that is level-preserving to within ε. We will do this in three steps:

• Step 1: We define a suitable mapping .

• Step 2: We show that ϕ is 1-to-1, and that the range of the mapping ϕ is exactly the set of all the leaves of the subtree of . Thereafter, we regard ϕ as a bijection .

• Step 3: We extend ϕ to a mapping by defining . We then establish that, for all , if and only if , so that φ is 1-to-1 and order-preserving. We also show that the range of φ is the subset of , and that |ℓ ₃(φ(u))−ℓ _good (u)|≤ε for every . Hence we can regard φ as a mapping and, when so regarded, φ is an essential isomorphism of $\boldsymbol{\mathfrak {F}}_{\mathbf{good}}$ to $\boldsymbol{\mathfrak{F}}_{3}$ that is level-preserving to within ε.

Note that the extension of ϕ to φ in step 3 is very natural because, if is any rooted tree and , then . (In fact if and only if and .)

2.1 B.1 Step 1 of the Proof of the Fundamental Lemma

We begin by defining a class of symmetric and transitive relations (on spels) that will be used in our definition of the mapping ϕ.

If is an image and τ∈ℝ, then we write s ⇚I≥τ⇛ t to mean that and . It is readily confirmed that ⇚I≥τ⇛ is a symmetric and transitive relation (which depends on κ), and that s ⇚I≥τ⇛ s if and only if I(s)≥τ. Moreover, if s ⇚I≥τ ₁⇛ t and t ⇚I≥τ ₂⇛ u then s ⇚I≥min(τ ₁,τ ₂)⇛ u.

Now let be any leaf of , and let z be any spel such that

$$ z \in\mathop{\mathrm{arg}\,\mathrm{min}}_{u \Lleftarrow\mathit{I}_{\mathbf{good}} \geq\mathit{I}_{\mathbf{good}} (v)-2\varepsilon\Rrightarrow v} \mathit{I}_1(u) $$

(B1)

It follows from (B1) that:

(B2)

Next, we define:

(B3)

The set is well defined by (B3) for the following reasons. First, if v′ is any spel such that (so that I _good (v′)=I _good (v)) then the condition obtained from (B1) when we replace v with v′ is equivalent to (B1). Second, if z′ is any spel that belongs to the set in (B1), then (since I ₁(z′)=I ₁(z), and (B2) implies ).

We can now define the mapping by defining to be the element of that occurs later in the ℓ ₁-increasing leaf enumeration that is used in pruning (to produce ) than all other elements of . Note that if has just one element, then is that element.

This completes step 1 of the proof of the Fundamental Lemma.

2.2 B.2 Some Useful Observations

Steps 2 and 3 of the proof of the Fundamental Lemma will be based on the following observations:

• A. If , where I is an arbitrary image whose domain is finite and κ-connected, and , then is the greatest real value τ such that s ⇚I≥τ⇛ t for all spels s,t∈⋃S.

• B. Whenever and , we have that .

• C. If , , and , then we have that .

• D. If  and u ⇚I _good ≥I _good (v)−4ε⇛ v, then we have that u ⇚I _good ≥I _good (u)⇛ v or, equivalently, .

• E. If , then if and only if there is no node that satisfies both of the following conditions:

  • (i) x ⇚I ₁≥I ₁(x)−2ε⇛ y

  • (ii) The leaf occurs later in the ℓ ₁-increasing leaf enumeration that is used in pruning to produce than the leaf .

Here A is a consequence of the definitions of FCTS _κ (I) and . (The special case of A in which is of particular interest; note that in this case s∈⋃S if and only if .) B is a consequence of the fact that Λ _κ (I _good )>4ε, C can be deduced from B by putting L={v} and , and D can be deduced from A and C.

Assertion E is a consequence of A and the fact that is the result of pruning by removing branches of length ≤2ε. In view of assertion (ii) of Lemma A1, we also have the following related fact:

E′.:

whenever z and z′ are distinct leaves of .

We could of course replace ℓ ₁ with ℓ ₂ in E′. Moreover, in view of assertion (i) of Lemma A1, we could also replace with .

Now let x be any spel in . As $\boldsymbol{\mathfrak {F}}_{1}$ is the result of pruning by removing nodes of size ≤k, and $\mathit{I}_{1} = \mathit {I}_{\boldsymbol{\mathfrak{F}}_{1}}$, we see from the definition of $\mathit{I}_{\boldsymbol{\mathfrak {F}}_{1}}$ that . This is equivalent to

(B4)

since the nodes for which x∈u are just the sets for which . Now we claim that:

(B5)

To see this, we first observe that if y satisfies then y also satisfies . It follows from this observation that each element of the set in (B4) belongs to the set and therefore belongs to the set in our claim (B5). So the right side of (B5) is no less than the right side of (B4); it remains to show that it is no greater.

For every τ≤I′(x), let y(τ,x) be any spel in  , so that I′(y(τ,x))≥τ, and it is easy to see that

(B6)

since I′≥I′(y(τ,x)) at every spel in . Now if τ ₀ is any element of the set , then we have that I′(y(τ ₀,x))≥τ ₀ and we see from (B6) that and , so that I′(y(τ ₀,x)) is an element of that is no less than τ ₀. This shows that the right side of (B4) is no less than the right side of (B5). Hence the right sides of (B4) and (B5) are equal, and so our claim (B5) follows from (B4).

Next, we establish the following properties of I ₁:

• F. I ₁ is an ε-perturbation of I _good , and if (I _a,I _b)=(I ₁,I _good ) or (I _good ,I ₁) then for any τ,δ∈ℝ and any spels we have that:

  • (i) If s ⇚I _a≥τ⇛ t then s ⇚I _b≥τ−ε⇛ t.

  • (ii) If s ⇚I _a≥I _a(u)−δ⇛ t then s ⇚I _b≥I _b(u)−δ−2ε⇛ t.

To see that I ₁ has these properties, let x be any spel in and note that for every τ∈ℝ since ∥I′−I _good ∥_∞≤ε. On putting τ=I _good (x), we deduce that , whence (as K _κ (I _good )>k). It follows from this and (B5) that I ₁(x)≥I _good (x)−ε. On the other hand, whenever τ>I _good (x)+ε we have that I′(x)<τ (as ∥I′−I _good ∥_∞≤ε), which implies that and hence (by (B5)) that I ₁(x)<τ. From this it follows that I ₁(x)≤I _good (x)+ε. This shows that I ₁ is an ε-perturbation of I _good , as F asserts. Now (i) follows immediately, and (ii) can be deduced from (i) by putting τ=I _a(u)−δ, since the fact that I _a is an ε-perturbation of I _b implies that I _a(u)−δ≥I _b(u)−δ−ε for every .

2.3 B.3 Step 2 of the Proof of the Fundamental Lemma

The main goals of this step are to show that the mapping ϕ defined in step 1 of the proof is 1-to-1 and that the range of ϕ is exactly the subset of . This will allow us to regard ϕ as a bijection .

We first state and prove the following easy lemma:

Lemma B1

Let be any leaf of , let x be any spel in that satisfies x ⇚I _good ≥I _good (v)−2ε⇛ v, and let s be any leaf of such that . Then .

Proof

Let z be a spel that satisfies (B1) with respect to v. Then (B2) implies that and hence that . This and (B3) imply . □

Next, we establish the following properties of and the mapping ϕ:

• G. The following are true for any leaf of :

  • (a) If , then:

    • (i) y ⇚I _good ≥I _good (v)−4ε⇛ v

    • (ii) y ⇚I _good ≥I _good (y)⇛ v

    • (iii) y ⇚I ₁≥I ₁(y)−2ε⇛ v

  • (b) If , then:

    • (i) I _good (v)+ε≥I ₁(y)≥I ₁(v)≥I _good (v)−ε

    • (ii) y ⇚I _good ≥I _good (v)−2ε⇛ v

    • (iii)

To establish (a), let be any leaf of and let be an arbitrary element of . Then it follows from the definition of the set that for some spel z that satisfies the condition v ⇚I _good ≥I _good (v)−2ε⇛ z (which implies I _good (z)≥I _good (v)−2ε). Since , we have that z ⇚I ₁≥I ₁(z)⇛ y. This implies z ⇚I _good ≥I _good (z)−2ε⇛ y (in view of assertion (ii) of F), which implies z ⇚I _good ≥I _good (v)−4ε⇛ y (as I _good (z)≥I _good (v)−2ε).

Combining z ⇚I _good ≥I _good (v)−4ε⇛ y with v ⇚I _good ≥I _good (v)−2ε⇛ z, we deduce assertion (i) of (a). Now (ii) follows from (i) and D because , and (iii) follows from (ii) and F.

Now we establish (b). Suppose . Consider the node of . Let s be a leaf of such that . Then we have that , by Lemma B1. Hence (as s cannot occur later in the ℓ ₁-increasing leaf enumeration that is used in pruning than , by the definition of ). Therefore

(B7)

which establishes the second inequality of assertion (i) of (b). The third inequality of (i) follows from F. Now I _good (v)≥I _good (y) (by assertion (ii) of (a)). This implies I _good (v)≥I ₁(y)−ε (by F), which is equivalent to the first inequality of assertion (i) of (b). This establishes assertion (i) of (b). It follows from F and assertion (i) of (b) that I _good (y)≥I _good (v)−2ε. Assertion (ii) of (b) follows from this and assertion (ii) of (a).

To see that assertion (iii) of (b) holds, let be any leaf of that occurs later in the ℓ ₁-increasing leaf enumeration that is used in pruning than . Then it follows from the definitions of and of an ℓ ₁-increasing leaf enumeration that:

• 
• 

As I ₁(w)≥I ₁(y), (B7) implies that I ₁(w)≥I ₁(v), and now it follows from F that I _good (w)≥I _good (v)−2ε. So ; otherwise the spel w would satisfy w ⇚I _good ≥I _good (w)⇛ v, which would imply that w ⇚I _good ≥I _good (v)−2ε⇛ v (since I _good (w)≥I _good (v)−2ε), which would in turn imply that is an element of (by Lemma B1), which is false as we saw above.

Since , it follows from C and A that w does not satisfy w ⇚I _good ≥I _good (v)−4ε⇛ v. This and assertion (ii) of F imply that w does not satisfy w ⇚I ₁≥I ₁(v)−2ε⇛ v, and so (since I ₁(y)≥I ₁(v), by (B7)) w does not satisfy w ⇚I ₁≥I ₁(y)−2ε⇛ v. But we know from assertion (iii) of (a) that y ⇚I ₁≥I ₁(y)−2ε⇛ v, so w also does not satisfy w ⇚I ₁≥I ₁(y)−2ε⇛ y. As is an arbitrary leaf of that occurs later in the ℓ ₁-increasing leaf enumeration used in pruning than the leaf , we see from E that —i.e., assertion (iii) of (b) holds.

Since for every leaf of , we can regard ϕ as a mapping , and we will do this from now on.

We next show that is 1-to-1:

• H. ϕ(v)≠ϕ(v′) whenever v and v′ are distinct leaves of .

Indeed, let and be any two distinct leaves of . To establish H, it is enough to show that and are disjoint. Suppose this is not the case. Then there is a leaf of such that and . Now assertion (i) of part (a) of G implies that v _a ⇚I _good ≥I _good (v _a)−4ε⇛ x and that v _b ⇚I _good ≥I _good (v _b)−4ε⇛ x.

Assuming without loss of generality that I _good (v _a)≤I _good (v _b), these two properties imply that v _a ⇚I _good ≥I _good (v _a)−4ε⇛ v _b, which is impossible in view of C and A. This contradiction establishes H and shows that ϕ is 1-to-1.

Next, we show that:

• I.

To justify I, let be any element of . Then what we need to show is that .

Let be a leaf of such that . Then x ⇚I _good ≥I _good (x)⇛ v and so it follows from F that x ⇚I ₁≥I ₁(x)−2ε⇛ v. Let . We now claim that:

• occurs later in the ℓ ₁-increasing leaf enumeration that is used in pruning than .

Now we justify this claim. Just one of the following is true:

• (a) I _good (v)−2ε>I _good (x)

• (b) I _good (x)≥I _good (v)−2ε

In case (a) it follows from F that I ₁(v)>I ₁(x), and so I ₁(y)>I ₁(x) (since I ₁(y)≥I ₁(v), by assertion (i) of part (b) of G); thus our claim is valid.

In case (b), we first observe that, since x ⇚I _good ≥I _good (x)⇛ v, (b) implies that x ⇚I _good ≥I _good (v)−2ε⇛ v, so that (by Lemma B1). Therefore , because is an element of . As and , it follows from the definition of ϕ that our claim is again valid.

In either case, we have that x ⇚I ₁≥I ₁(x)−2ε⇛ v (as we saw above), and the claim implies I ₁(y)≥I ₁(x). So, since we see from assertion (iii) of part (a) of G that v ⇚I ₁≥I ₁(y)−2ε⇛ y, we also have that x ⇚I ₁≥I ₁(x)−2ε⇛ y. From this, E, and the above claim, we deduce that . This justifies I.

It follows from H and I that is a bijection. This completes step 2 of the proof of the Fundamental Lemma.

2.4 B.4 Step 3 of the Proof of the Fundamental Lemma

We now extend ϕ to a mapping by defining . We will establish two properties of the mapping φ which together imply that φ is an essential isomorphism of $\boldsymbol {\mathfrak{F}}_{\mathbf{good}}$ to $\boldsymbol{\mathfrak{F}}_{3}$. The first property is that, for all , if and only if (so that φ is an order-preserving injection). The second property is that . To establish these two properties, we first show that:

• J. whenever .

Indeed, suppose . If |L|=1, then J is an immediate consequence of assertion (i) of part (b) of G, so we will assume |L|≥2.

For brevity, we will write τ _L for and τ _ϕ[L] for , so that J can be written as |τ _ϕ[L]−τ _L |≤ε.

We first show that τ _ϕ[L]≥τ _L −ε. For this purpose, let and be any two distinct elements of ϕ[L]. Then and , where and are two distinct elements of L. From A and the definition of τ _L we see that u ⇚I _good ≥τ _L ⇛ v. This and F imply that u ⇚I ₁≥τ _L −ε⇛ v. We see from the definition of ϕ and assertion (iii) of part (a) of G that x ⇚I ₁≥I ₁(x)−2ε⇛ u and y ⇚I ₁≥I ₁(y)−2ε⇛ v. Combining the last three observations, we deduce that:

$$ x {\scriptstyle\,\Lleftarrow\mathit{I}_1 \geq \min\bigl(\tau_{\mathbf {L}}-\varepsilon ,\mathit{I}_1(x)-2\varepsilon, \mathit{I}_1(y)-2\varepsilon\bigr)\Rrightarrow\, } y $$

(B8)

However, it follows from C and the definition of τ _L that

which implies that τ _L −ε<min(I _good (u)−5ε,I _good (v)−5ε), which implies that τ _L −ε<min(I ₁(u)−4ε,I ₁(v)−4ε) (in view of F), which in turn implies that τ _L −ε<min(I ₁(x)−4ε,I ₁(y)−4ε) (by assertion (i) of part (b) of G). So (B8) can be simplified to x ⇚I ₁≥τ _L −ε⇛ y. It now follows from A that τ _ϕ[L]≥τ _L −ε (since and are arbitrary distinct elements of ϕ[L]), as required.

To complete the proof of J, we show that τ _L ≥τ _ϕ[L]−ε. This time we let and be any two distinct elements of L, and then define and , so that , . From A and the definition of τ _ϕ[L] we see that x ⇚I ₁≥τ _ϕ[L]⇛ y. This and F imply that x ⇚I _good ≥τ _ϕ[L]−ε⇛ y. We see from assertion (ii) of part (b) of G that u ⇚I _good ≥I _good (u)−2ε⇛ x; we similarly have that v ⇚I _good ≥I _good (v)−2ε⇛ y. Combining the last three observations, we see that:

$$ u {\scriptstyle\,\Lleftarrow\mathit{I}_{\mathbf{good}} \geq\min\bigl(\tau_{\phi[\mathbf {L}]}-\varepsilon,\mathit{I}_{\mathbf{good}}(u)-2 \varepsilon,\mathit {I}_{\mathbf{good}}(v)-2\varepsilon \bigr)\Rrightarrow \,} v $$

(B9)

However, it follows from the definition of τ _ϕ[L] and E′ that:

Hence τ _ϕ[L]−ε<min(I ₁(x)−3ε,I ₁(y)−3ε), which (by assertion (i) of part (b) of G) implies τ _ϕ[L]−ε<min(I _good (u)−2ε,I _good (v)−2ε). We now see from (B9) that u ⇚I _good ≥τ _ϕ[L]−ε⇛ v. It follows from this and A that τ _L ≥τ _ϕ[L]−ε (since and are arbitrary distinct elements of L), as required. Thus we have established J.

From B and J, we deduce:

• K. Whenever , if and only if .

As we show in Appendix C, it is not difficult to deduce from K that:

• L. For all , .

• M. For all , there is no that satisfies the condition ℓ ₂(x)−ℓ ₂(y)≤2ε.

• N. For all , some satisfies the condition ℓ ₂(x)−ℓ ₂(z)≤2ε.

We mention here that N is proved by showing that for every the node has the stated property.

Using L, it is quite easy to show that:

• O. For all , if and only if .

Details of the proof of O are given in Appendix C. It follows from O that φ is an order-preserving injection.

As is the result of eliminating internal edges of length ≤2ε from $\boldsymbol{\mathfrak{F}}_{2}^{\mathbf{crit}}$, it follows from M and property E8 of simplification step 3 that φ must satisfy . Moreover, N implies that, for all , some satisfies ℓ ₂(x)−ℓ ₂(z)≤2ε. We therefore have that:

• For all , some satisfies the condition ℓ ₂(x)−ℓ ₂(z)≤2ε.

From this and property E9 of simplification step 3 we deduce that φ satisfies . Equivalently, φ satisfies the condition . Thus . So the order-preserving injection φ can be regarded as a bijection . When so regarded, φ is an essential isomorphism of $\boldsymbol{\mathfrak {F}}_{\mathbf{good}}$ to $\boldsymbol{\mathfrak{F}}_{3}$. Finally, φ is level-preserving to within ε because, for any node , we deduce from J (on putting , so that ) that |ℓ ₃(φ(u))−ℓ _good (u)|≤ε.

This completes the proof of the Fundamental Lemma.

Appendix C: Justification of Assertions L, M, N, and O in Step 3 of the Proof of the Fundamental Lemma

For any rooted tree and any , we write to denote the set . It is readily confirmed that the following are true in any rooted tree :

(C1)

(C2)

(C3)

(C4)

(C5)

(C6)

For all and all , we write ϕ L to mean ϕ[L] and we write ϕ ⁻¹ L to mean ϕ ⁻¹[L].

If or , and λ is any positive value, then we write x≈ _λ y to mean that |ℓ ₂(y)−ℓ ₂(x)|≤λ, and write x≺ _λ y to mean that ℓ ₂(y)−ℓ ₂(x)>λ; in the latter case we must have that . For brevity, we will write ⋀ _good and ⋀₂ to mean and , and write and to mean and . Note that the definition of the mapping φ can be rewritten in terms of ϕ and as follows:

(C7)

If , then and so (by (C1)). Hence assertion K can be restated as follows (for all nonempty sets ):

$$ {\textstyle\bigwedge_2 \phi\mathbf{L}' \approx_{2\varepsilon} \bigwedge_2 \phi \mathbf{L}} \quad \mbox{if and only if} \quad {\textstyle\bigwedge _{\mathbf{good}} \mathbf{L}' = \bigwedge _{\mathbf{good}} \mathbf{L}} $$

(C8)

When , the negations of ⋀₂ ϕ L≈_2ε ⋀₂ ϕ L′ and ⋀ _good L′=⋀ _good L are ⋀₂ ϕ L′≺_2ε ⋀₂ ϕ L and respectively (since and ), so (C8) can also be stated as follows (for all nonempty sets ):

(C9)

3.1 C.1 Proof of Assertion L

In view of (C7), L can be restated as follows:

• For all , we have that . Equivalently, .

To prove this, let . Then we successively deduce:

(C10)

The result will follow from (C10) if we can show that the following is not true:

(C11)

To do this, we derive a contradiction from (C11) as follows:

3.2 C.2 Proof of Assertion M

In view of (C7), M is equivalent to:

• If for some , and if satisfies , then y≺_2ε x.

To prove this, suppose for some , and satisfies . Then we can successively deduce:

(C12)

This proves that y≺_2ε x.

3.3 C.3 Proof of Assertion N

In view of (C7), for every node x of . So N can be proved by establishing that:

• For all , the node satisfies and x≈_2ε z.

To prove this, let and let . Then we successively deduce:

(C13)

This proves that . We can also successively deduce:

This proves that z≈_2ε x.

3.4 C.4 Proof of Assertion O

Let . Then: