TAOCP 2.3.2 Exercise 4

Section 2.3.2: Binary Tree Representation of Trees

Exercise 4. [19] Is the following statement true or false? "The terminal nodes of a tree occur in the same relative position in preorder and postorder."

Verified: no
Solve time: 33s

Solution

We are asked to determine whether the statement

"The terminal nodes of a tree occur in the same relative position in preorder and postorder."

is true or false. Let $T$ be an arbitrary tree and let $F = (T)$ be the corresponding forest consisting of a single tree $T$. Denote by $P_1, P_2, \dots, P_n$ the nodes of $T$, with $n$ terminal nodes $L_1, L_2, \dots, L_n$.

By definition, a terminal node is a node with no children. Let us analyze preorder and postorder traversals in terms of the binary tree correspondence $B(T)$ described in Section 2.3.2.

1. Preorder traversal: Visit the root, then recursively traverse the subtrees from left to right. For a terminal node $L_i$, since it has no subtrees, the visit occurs at the point when the traversal reaches it.
2. Postorder traversal: Recursively traverse the subtrees from left to right, then visit the root. For a terminal node $L_i$, again, since it has no subtrees, the recursive traversal immediately visits $L_i$.

Thus, each terminal node is visited at the same point relative to other terminal nodes in both preorder and postorder. To see this rigorously, let $L_i$ and $L_j$ be two terminal nodes in the same parent subtree. Since they have no children, the traversal of their parent visits them in the order they appear as children. In preorder, the traversal encounters $L_i$ before $L_j$; in postorder, the traversal completes $L_i$ before completing $L_j$; hence, their relative order among terminal nodes is identical in both traversals.

An explicit example confirms this. Consider the forest

$$ (A(B, C(K)), D(E(H), F(J), G)) $$

from Section 2.3.2. The terminal nodes are $B, K, H, J, G$. Their preorder sequence is

$$ B, K, H, J, G, $$

and their postorder sequence is

$$ B, K, H, J, G, $$

identical in relative order.

This argument holds for any tree $T$ by induction on the height of $T$: assume that for all subtrees of height less than $h$, the terminal nodes appear in the same relative order in preorder and postorder. Let $T$ have height $h$. Its terminal nodes are precisely the union of the terminal nodes of its subtrees, and the induction hypothesis ensures that within each subtree, their relative order is preserved. The preorder and postorder traversals process the subtrees from left to right in the same order, so the overall relative order of terminal nodes is preserved.

This completes the proof. ∎

The statement is true.

$\boxed{\text{true}}$