Society of Actuaries (SOA) PA Practice Exam

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Which characteristic defines an unbalanced binary tree?

Both subtrees are of equal depth
Leaves are at different levels
Subtrees differ in depth by more than one
It has a maximum of two children per node

The correct answer is: Subtrees differ in depth by more than one

An unbalanced binary tree is characterized by having subtrees that differ in depth by more than one. In a balanced binary tree, the depth of the left and right subtrees for any node must not differ by more than one, ensuring that the tree remains efficiently structured for operations like insertion, deletion, and search. When this balance is disrupted, specifically when the difference in depth exceeds one, the tree is classified as unbalanced. In contrast, the other characteristics described do not specifically capture the essence of an unbalanced binary tree. For instance, having leaves at different levels can occur in a balanced tree as well; the balance condition primarily focuses on the depth difference between immediate child subtrees rather than the leaves themselves. Furthermore, even if both subtrees are of equal depth, the tree can still be considered balanced. The restriction of having a maximum of two children per node is a defining characteristic of all binary trees, balanced or unbalanced, and does not pertain specifically to the concept of balance in the tree structure. Thus, the defining feature of an unbalanced binary tree is that its subtrees differ in depth by more than one.