Society of Actuaries (SOA) PA Practice Exam

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What distinguishes a balanced binary tree?

Subtrees differ in depth by at most two
Both subtrees of every node differ in depth by at most one
All nodes must have exactly two children
It must have at least one leaf node

The correct answer is: Both subtrees of every node differ in depth by at most one

A balanced binary tree is characterized by the condition that both subtrees of every node differ in depth by at most one. This definition ensures that the tree maintains a relatively even structure, which is crucial for efficiency in various operations such as insertion, deletion, and searching. In a balanced binary tree, the height difference between the left and right subtrees for any given node must not exceed one. This balance contributes to ensuring that the operations maintain a logarithmic time complexity relative to the number of nodes in the tree. Therefore, when trees remain balanced, they minimize the potential for becoming skewed, which could degrade performance. The other definitions do not correctly apply to the standard concept of balanced binary trees. A situation where subtrees differ in depth by at most two does not guarantee balance and may lead to scenarios that could adversely affect performance. Requiring all nodes to have exactly two children does not reflect the flexibility allowed in binary trees, as they can have one or two children at any node. Lastly, having at least one leaf node is a superficial condition that does not influence the balance of the tree structure.