Planar upward tree drawings with optimal area

Rooted trees are usually drawn planar and upward i e without crossings and with out any parent placed below its child In this paper we investigate the area requirement of planar upward drawings of rooted trees We give tight upper and lower bounds on the area of various types of drawings and provide linear time algorithms for constructing optimal area drawings Let T be a bounded degree rooted tree with N nodes Our results are summarized as follows We show that T admits a planar polyline upward grid drawing with area O N and with width O N for any prespeci ed constant such that If T is a binary tree we show that T admits a planar orthogonal upward grid drawing with area O N log logN We show that if T is ordered it admits an O N logN area planar upward grid drawing that preserves the left to right ordering of the children of each node We show that all of the above area bounds are asymptotically optimal in the worst case We present O N time algorithms for constructing each of the above types of drawings of T with asymptotically optimal area We report on the experimentation of our algorithm for constructing planar poly line upward grid drawings performed on trees with up to million nodes