#### Binary Trees

```/* insert_leaf(X, Tree, Tree1) is true if Tree1 is the result of inserting */
/*   the element X as a leaf in the ordered binary tree Tree.  If the      */
/*   element X is already in the tree, the tree is unchanged.              */
insert_leaf(X, void, tree(X,void,void)):-!.
insert_leaf(X, Tree, Tree):-
Tree=tree(X,_,_), !.
insert_leaf(X, tree(Y,L,R), tree(Y,L1,R)):-
X < Y, !,
insert_leaf(X, L, L1).
insert_leaf(X, tree(Y,L,R), tree(Y,L,R1)):-
insert_leaf(X, R, R1).

/* delete_node(X, Tree, Tree1) is true if Tree1 is the result of deleting  */
/*   the element X from the ordered binary tree Tree.                      */
delete_node(X, tree(X,L,void), L):-!.
delete_node(X, tree(X,L,R), Tree):-!,
left_rest(R, Y, R1),
Tree=tree(Y,L,R1).
delete_node(X, tree(Y,L,R), tree(Y,L1,R)):-
X < Y, !,
delete_node(X, L, L1).
delete_node(X, tree(Y,L,R), tree(Y,L,R1)):-
delete_node(X, R, R1).

/* left_rest(Tree, Left, Rest) is true if Left is the leftmost element in  */
/*   the binary tree Tree, and Rest is the rest of the tree.               */
left_rest(tree(X,void,R), X, R):-!.
left_rest(tree(X,L,R), Y, tree(X,L1,R)):-left_rest(L, Y, L1).

/* pre_order(Tree, L) is true if L is a pre-order traversal of the binary  */
/*   tree Tree.                                                            */
pre_order(T, L):-pre_order_dl(T, L, []).

pre_order_dl(tree(X,L,R), [X|Xs], Zs):-
pre_order_dl(R, Ys, Zs),
pre_order_dl(L, Xs, Ys).
pre_order_dl(void, Xs, Xs).

/* in_order(Tree, L) is true if L is an in-order traversal of the binary   */
/*   tree Tree.                                                            */
in_order(T, L):-in_order_dl(T, L, []).

in_order_dl(tree(X,L,R), Xs, Zs):-
in_order_dl(R, Ys, Zs),
in_order_dl(L, Xs, [X|Ys]).
in_order_dl(void, Xs, Xs).

/* post_order(Tree, L) is true if L is a post-order traversal of the       */
/*   binary tree Tree.                                                     */
post_order(T, L):-post_order_dl(T, L, []).

post_order_dl(tree(X,L,R), Xs, Zs):-
post_order_dl(R, Ys, [X|Zs]),
post_order_dl(L, Xs, Ys).
post_order_dl(void, Xs, Xs).

/* breadth_order(BinaryTree, List) is true if List is the level-by-level   */
/*   traversal of BinaryTree.                                              */
/* This procedure uses a queue implemented as a difference list with a     */
/*   counter. It can be used backwards, that is, it can be used to         */
/*   enumerate, by backtracking, every BinaryTree for which List is the    */
/*   level-by-level traversal.  For a list of n elements, the number of    */
/*   binary trees is the n-th Catalan number - see "Data Structures and    */
/*   Program Design in C" by Kruse, Leung and Tondo.                       */