properties of binary trees binary trees have several interesting properties dealing with relationships between their heights and number of nodes we denote the set of all nodes of tree at the same depth as level of in binary treelevel has at most one node (the root)level has at most two nodes (the children of the root)level has at most four nodesand so on (see figure in generallevel has at most nodes nodes level figure maximum number of nodes in the levels of binary tree we can see that the maximum number of nodes on the levels of binary tree grows exponentially as we go down the tree from this simple observationwe can derive the following properties relating the height of binary tree with its number of nodes detailed justification of these properties is left as exercise - proposition let be nonempty binary treeand let nne ni and denote the number of nodesnumber of external nodesnumber of internal nodesand height of respectively then has the following properties < < + <ne < <ni < log( < < alsoif is properthen has the following properties < < + <ne < <ni < log( < <( )/