finiteheight

Finiteheight is a term used in mathematics and computer science to describe a property of hierarchical structures where the maximal degree of vertical extension is finite. It is defined via the concept of height, which depends on the chosen base element: for rooted trees, the height is the length of the longest path from the root to a leaf; for partially ordered sets, the height is the supremum of the lengths of all chains.

In rooted trees, finiteheight means there exists a finite bound h such that no root-to-leaf path exceeds h edges. Finiteheight trees include all finite trees; an infinite tree can also have finite height if its paths from the root are all of length at most h while the tree has unbounded breadth. The height is sometimes called the depth of the tree.

In posets, finite height means there is a finite bound on the length of chains; equivalently, there is no infinite strictly increasing sequence. Finite height posets allow certain decompositions and simplifications in order-theoretic arguments.

Examples: a finite binary tree has finite height equal to the maximum depth; an infinite binary tree generally has infinite height if every branch is unbounded.

Applications include analysis of algorithmic complexity, where operation times can be bounded by height in certain tree-based data structures, and structural theory in order theory, where height informs chain decompositions and related properties.

Computation: for rooted trees, height is computed by a depth-first traversal or by recursive depth; for posets, one can model the Hasse diagram as a DAG and compute the longest path length by dynamic programming in O(n+m) time.