Algorithms and Data Structures question Let T be a proper binary tree. Given a n

Question

Algorithms and Data Structures question

Let T be a proper binary tree. Given a node v E T, the imbalance of v denoted imbalance(v), is defined as the difference in absolute value, between the number of leaves of the left subtree of v and the number of leaves of the right subtree of v. (If v is a leaf, then imbalance(v) is defined to be 0.) Define imbalance(TT) J VET imbalance(v). (a) Provide an upper bound to the imbalance of a proper binary tree with n nodes, and exhibit a proper binary tree whose imbalance matches such an upper bound. (b) Draw a proper binary tree T for which imbalance (T) imbalance v and v is not the root of T

Explanation / Answer

(a) In a completely balanced binary tree, the following property holds for every node: The number of nodes in its left subtree and the number of nodes in its right subtree are almost equal, which means their difference is not greater than one.

Hence, Imbalance of a node(v) = abs(no. of nodes in its left subtree - no. of nodes in its right tree)

Note: Here, abs means absolute value.

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