c++ new operator error

This is because there is a problem with the constructor declaration of binary_tree_node.

If you look at the constructor part in the code you showed me, it is as follows.

binary_tree_node();
binary_tree_node<T>(const binary_tree_node& copy);
binary_tree_node<T>(const T& init_data = T(), binary_tree_node* init_left = NULL, binary_tree_node* init_right = NULL);

If you look at the third constructor, the default values are applied to all call factors.

binary_tree_node<int>()

This could be the first constructor, or the third constructor with all the default values applied.

To resolve the problem, remove the default values for the init_data variable as shown below.

binary_tree_node<T>(const T& init_data, binary_tree_node* init_left = NULL, binary_tree_node* init_right = NULL);