- let be (not necessarily properbinary tree with nodesand let be the sum of the depths of all the external nodes of show that if has the minimum number of external nodes possiblethen is (nand if has the maximum number of external nodes possiblethen is ( log nc- let be (possibly improperbinary tree with nodesand let be the sum of the depths of all the external nodes of describe configuration for such that is ( such tree would be the worst case for the asymptotic running time of method height (code fragment - for tree let ni denote the number of its internal nodesand let ne denote the number of its external nodes show that if every internal node in has exactly childrenthen ne ni - two ordered trees and are said to be isomorphic if one of the following holdsboth and are empty the roots of and have the same number > of subtreesand the ith such subtree of is isomorphic to the ith such subtree of for design an algorithm that tests whether two given ordered trees are isomorphic what is the running time of your algorithmc- show that there are more than improper binary trees with internal nodes such that no pair are isomorphic (see exercise - - if we exclude isomorphic trees (see exercise - )exactly how many proper binary trees exist with exactly leavesc- add support in linkedbinarytree for methoddelete subtree( )that removes the entire subtree rooted at position pmaking sure to maintain the count on the size of the tree what is the running time of your implementationc- add support in linkedbinarytree for methodswap( , )that has the effect of restructuring the tree so that the node referenced by takes the place of the node referenced by qand vice versa make sure to properly handle the case when the nodes are adjacent - we can simplify parts of our linkedbinarytree implementation if we make use of of single sentinel nodereferenced as the sentinel member of the tree instancesuch that the sentinel is the parent of the real root of the treeand the root is referenced as the left child of the sentinel furthermorethe sentinel will take the place of none as the value of the left or right member for node without such child give new implementation of the update methods delete and attachassuming such representation