Height, Level and Depth of a Tree

DSA, Database System & Operating System — Data Structure, Lists, Linked Lists and Trees, NEC licence examination syllabus (Nepal Engineering Council).

Height, Level, and Depth of a Tree

Three closely related terms that examiners love mixing up on purpose.

📏 The Three Measurements

Depth (of a node)Number of edges from the ROOT down to that node.

Height (of a node)Number of edges from that node DOWN to its deepest leaf.

LevelDepth + 1 (the root is commonly called level 1, though some texts start at 0 — always check your textbook's convention).

💡 One-liner: "What is the height of the tree overall?" → The height of its ROOT node — the longest path from root to any leaf.

The distinction that decides the answer

Depth counts upward to the root and belongs to a node. Height counts downward to the deepest leaf. Both are properties of a node — but "the height of the tree" conventionally means the height of the root.

The consequence worth internalising: every leaf has height 0, and the root has depth 0. If a question's numbers seem off by one, it is almost always because depth and height were swapped, or because a source counts nodes on the path rather than edges.

💡 State your convention in the exam. Counting edges is the more common convention and gives a single node a height of 0; counting nodes gives it 1. Both appear in textbooks, so writing "counting edges, height = 3" earns the mark either way, while a bare "3" is at the examiner's mercy.

Why height is the number that matters

Every BST operation costs the height of the tree, not the number of nodes — so height is the performance. For n nodes the height can be anywhere between about log₂n when perfectly balanced and n−1 when fully degenerate.

That range is the entire justification for AVL and other self-balancing trees: they do extra work on insertion specifically to keep the height near log₂n, buying a guaranteed bound instead of hoping the input arrives in a lucky order.

Syllabus points

Definitions and computation

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