# Minimum-area h-v drawings of complete binary trees

## Abstract

We study the area requirement of h-v drawings of complete binary trees. An h-v drawing of a binary tree *t* is a drawing of *t* such that (a) nodes are points with integer coordinates, (b) each edge is either a rightward-horizontal or a downward-vertical straight-line segment from a node to one of its children, (c) edges do not intersect, and (d) if *t*_{1} and *t*_{2} are immediate subtrees of a node *u*, the enclosing rectangles of the drawings of *t*_{1} and *t*_{2} are disjoint. We prove that, for any complete binary tree *t* of height *h* ≥ 3 and with *n* nodes, the area of the optimum h-v drawing of *t* is equal to (a) 2.5*n* − 4.5 \(\sqrt {(n + 1)/2} \)+ 3.5 if *h* is odd, (b) 2.5*n* − 3.25 \(\sqrt {n + 1} \)+ 3.5 otherwise. As far as we know, this is one of the few examples in which a closed formula for the minimum-area drawing of a graph has been explicitly found. Furthermore this minimum-area h-v drawing can be constructed in linear time. As a consequence of this result and the result of Trevisan (1996), we have that h-v drawings are *provably* less area-efficient than strictly upward drawings when we restrict ourselves to complete binary trees. We also give analogous results for the minimum-perimeter and the minimum-enclosing square area h-v drawings.

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