GD 1997: Graph Drawing pp 371-382

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

Extended abstract
• P. Crescenzi
• P. Penna
Packing, Compressing, and Touching
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1353)

## 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 t1 and t2 are immediate subtrees of a node u, the enclosing rectangles of the drawings of t1 and t2 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) 2.5n − 4.5 $$\sqrt {(n + 1)/2}$$+ 3.5 if h is odd, (b) 2.5n − 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.

## References

1. 1.
T. Chan, M. T. Goodrich, S. R. Kosaraju, and R. Tamassia. Optimizing Area and Aspect Ratio in Straight-Line Orthogonal Tree Drawings. In Proc. Graph Drawing 96, LNCS 1190, 63–75, 1997.Google Scholar
2. 2.
P. Crescenzi, G. Di Battista, and A. Piperno. A note on optimal area algorithms for upward drawings of binary trees. Computational Geometry: Theory and Applications, 2:187–200, 1992.
3. 3.
P. Crescenzi and P. Penna. Upward Drawings of Search Trees. In Proc. WG 96, LNCS 1197, 114–125, 1997.Google Scholar
4. 4.
P. Crescenzi, P. Penna, and A. Piperno. Linear area upward drawings of AVL trees. Computational Geometry: Theory and Applications, to appear.Google Scholar
5. 5.
P. Crescenzi, and A. Piperno. Optimal-area upward drawings of AVL trees. In Proc. Graph Drawing 94, LNCS 894, 307–317, 1994.Google Scholar
6. 6.
P. Eades, T. Lin, and X. Lin. Minimum size h-v drawings. In Advanced Visual Interfaces, 386–394, World Scientific, 1992.Google Scholar
7. 7.
A. Garg, M. T. Goodrich, and R. Tamassia. Planar upward tree drawings with optimal area. IJCGA, 6:333–356, 1996.
8. 8.
P.T. Metaxas, G.E. Pantziou, and A. Symvonis. Parallel h-v drawings of Binary Trees. In Proc ISAAC 94, 487–496, 1994.Google Scholar
9. 9.
Y. Shiloach. Linear and planar arrangements of graphs. Ph.D. Thesis, Department of Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel, 1976.Google Scholar
10. 10.
C.-S. Shin, S.K. Kim, and K.-Y. Chwa. Area-Efficient Algorithms for Upward Straight-Line Tree Drawings. In Proc. COCOON 96, LNCS 1090, 106–116, 1996.Google Scholar
11. 11.
L. Trevisan. A Note on Minimum-Area Upward Drawing of Complete and Fibonacci Trees. Information Processing Letters, 57:231–236, 1996.