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**Maximal path length of binary trees.**
*(English)*
Zbl 0821.68094

Summary: We further refine the bounds on the path length of binary trees of a given size by considering not only their sizes, but also their heights and fringe thicknesses (the difference between the length of their shortest root-to-leaf paths and their heights). We characterize the maximum-path-length binary trees of a given height, size and fringe thickness, and using this characterization, we give an algorithm to find the maximum-path-length binary trees of a given size and fringe thickness. The proof of the main result is based on two new tree transformations that preserve the height, size, and fringe thickness.

### Keywords:

path length of binary trees
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\textit{H. Cameron} and \textit{D. Wood}, Discrete Appl. Math. 55, No. 1, 15--35 (1994; Zbl 0821.68094)

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### References:

[1] | Cameron, H., Extremal cost binary trees, Ph.D. thesis, (1991), University of Waterloo |

[2] | Cameron, H.; Wood, D., Pm numbers, ambiguity, and regularity, RAIRO informatique théorique, 27, 261-275, (1993) · Zbl 0806.11007 |

[3] | Cameron, H.; Wood, D., Binary trees, fringe thickness, and minimum path length, (1993), unpublished manuscript |

[4] | De Santis, A.; Persiano, G., Tight upper and lower bounds on the path length of binary trees, (1991), unpublished manuscript · Zbl 0767.05060 |

[5] | Knuth, D.E., The art of computer programming, vol. 3: sorting and searching, (1973), Addison-Wesley Reading, MA · Zbl 0302.68010 |

[6] | Klein, R.; Wood, D., On the path length of binary tree, J. ACM, 36, 280-289, (1989) · Zbl 0674.68012 |

[7] | Nievergelt, J.; Wong, C.-K., Upper bounds for the total path length of binary trees, J. ACM, 20, 1-6, (1973) · Zbl 0263.68022 |

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