lu(qt)which returns the left-upper quad-tree ru(qt)which returns the right-upper quad-tree ll(qt)which returns the left-lower quad-tree rl(qt)which returns the right-lower quad-tree which can be applied whenever isvalue(qtis false for cases when isvalue(qtis truewe could define an operator value(qtthat returns the valuebut conventionally we simply say that qt itself is the required value quad-trees of this type are most commonly used to store grey-value pictures (with representing black and whitea simple example would be we can then create algorithms using the operators to perform useful manipulations of the representation for examplewe could rotate picture qt by usingrotate(qtif isvalue(qtreturn qt else return makeqtrotate(rl(qt))rotate(ll(qt))rotate(ru(qt))rotate(lu(qt)or we could compute average values by recursively averaging the constituent sub-trees there exist numerous variations of this general ideasuch coloured quadtrees which store value-triples that represent colours rather than grey-scaleand edge quad-trees which store lines and allow curves to be represented with arbitrary precision binary trees binary trees are the most common type of tree used in computer science binary tree is tree in which every node has at most two childrenand can be defined "inductivelyby the following rules