binary trees (where the character indicates integer division with no remainderyou can check this out by looking at figure array null null null null null figure tree represented by an array in most situationsrepresenting tree with an array isn' very efficient unfilled nodes and deleted nodes leave holes in the arraywasting memory even worsewhen deletion of node involves moving subtreesevery node in the subtree must be moved to its new location in the arraywhich is time-consuming in large trees howeverif deletions aren' allowedthe array representation may be usefulespecially if obtaining memory for each node dynamically isfor some reasontoo timeconsuming the array representation may also be useful in special situations the tree in the workshop appletfor exampleis represented internally as an array to make it easy to map the nodes from the array to fixed locations on the screen display duplicate keys as in other data structuresthe problem of duplicate keys must be addressed in the code shown for insert()and in the workshop appleta node with duplicate key will be inserted as the right child of its twin the problem is that the find(routine will find only the first of two (or moreduplicate nodes the find(routine could be modified to check an additional data itemto distinguish data items even when the keys were the samebut this would be (at least somewhattime-consuming