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Recipe

Find parent, children, level, and height in a rooted tree

Use when: you're given a rooted tree and asked to name a vertex's parent, children, or ancestors, its level, or the tree's height

Goal: every relationship and level value read directly off the tree, with no guessing

Taught in Rooted trees

Quick recipe

  1. Trace the one path from the vertex back to the root
  2. Read off the parent and every ancestor from that path
  3. Identify children by looking one level down
  4. Count edges on the root path to get the level
  5. Find the tree's height by checking every root-to-leaf path
  1. 1

    Follow tree edges upward from the target vertex until reaching the root. A rooted tree guarantees exactly one such path.

    Example

    For vertex D in the labeled tree (root R; children A, B, C; A's children D, E; C's child F), the path back to the root is D \to A \to R.

    Why: every vertex in a rooted tree has a unique path to the root, so this path always exists and is never ambiguous.

  2. 2

    The vertex immediately above the target on the path is its parent. Every other vertex on the path is an ancestor.

    Example

    On the path D \to A \to R, A sits directly above D, so A is D's parent, and both A and R are D's ancestors.

    • the target vertex is the root itself — it has no parent and no ancestors, and its level is 0.
    • the target vertex is any other vertex — it always has exactly one parent, the vertex found in Step 1.

    Why: parent names the one vertex directly above the target, while ancestor extends that relationship to every vertex farther up the same path.

  3. 3

    A vertex's children are the neighbors whose own path to the root passes through it. In a drawn tree, these sit directly below it, joined by an edge.

    Example

    A's neighbors below it are D and E, so A's children are D and E.

    Why: a child is exactly a neighbor farther from the root, the reverse of the parent relationship.

  4. 4

    The level of a vertex is the number of edges on its unique path back to the root, that is, the path's length.

    Example

    A's path to the root is A \to R, one edge, so A sits at level 1. D's path is D \to A \to R, two edges, so D sits at level 2.

    Why: level measures distance from the root, and distance in a tree is exactly the number of edges on the one connecting path.

  5. 5

    List every path from the root to a leaf, count the edges on each, and take the largest count. That largest value is the height.

    Example

    R \to B has length 1. R \to A \to D and R \to A \to E each have length 2. R \to C \to F also has length 2, so the largest is 2.

    Why: height is defined as the largest level any vertex reaches, so it is a maximum over every root-to-leaf path, not an average.

Common mistakes

  • Naming an ancestor two or more steps up the path as if it were the parent.
  • Computing height by averaging path lengths instead of taking the largest one.
  • Confusing a vertex's children with its full set of descendants further down the tree.
  • Miscounting level by counting the vertices on the path instead of the edges between them.