CREATE OR REPLACE FUNCTION tree_child(integer[2], integer)
RETURNS integer[2] AS '
DECLARE
	node ALIAS FOR $1;
	child ALIAS FOR $2;
BEGIN
	RETURN ARRAY[
		node[1] * (2 << child) + 3 - (2 << child),
		node[2] * (2 << child)
	];
END;
' LANGUAGE plpgsql
IMMUTABLE
RETURNS NULL ON NULL INPUT;

CREATE OR REPLACE FUNCTION tree_parent(integer[2])
RETURNS integer[2] AS '
DECLARE
	node ALIAS FOR $1;
	parent integer[2];
BEGIN
	IF node[1] = 3 THEN RETURN NULL; END IF;
	parent := ARRAY[
		(node[1]-1) >> 1, 
		node[2] >> 1
	];
	WHILE (parent[1]-1) % 4 = 0 LOOP
		parent := ARRAY[
			(parent[1]+1) >> 1,
			parent[2] >> 1
		];
	END LOOP;
	RETURN parent;
END;
' LANGUAGE plpgsql
IMMUTABLE
RETURNS NULL ON NULL INPUT;

CREATE OR REPLACE FUNCTION tree_distance(integer[2], integer[2])
RETURNS integer AS '
DECLARE
	child ALIAS FOR $1;
	parent ALIAS FOR $2;
BEGIN
	IF child IS NULL THEN RETURN NULL; END IF;
	IF child[1] = parent[1] AND child[2] = parent[2] THEN RETURN 0; END IF;
	RETURN 1 + tree_distance( tree_parent(child), parent );
END;
' LANGUAGE plpgsql
IMMUTABLE
RETURNS NULL ON NULL INPUT;