"On the shape of binary trees" Mireille Bousquet-Mélou, CNRS, LaBRI Université Bordeaux 1 Take a random binary tree with n nodes, and embed it in Z² in a canonical way: that is, the root of the tree sits at the origin (0,0), and each right [left] son of a node lies one unit above and one unit to the right [left] of its father. Using techniques from enumerative combinatorics and complex analysis, we shall study the typical shape of this canonical embedding for large binary trees of a given size. This includes questions like: how high, how wide is the embedding? What are the horizontal and vertical profiles of the tree? That is, how many nodes lie at a given abscissa or ordinate? The results dealing with the horizontal profile are not new, while those dealing with the vertical profile are recent (http://fr.arxiv.org/abs/math.CO/0501266). Their main motivation lies in the connection between embeddings of binary trees and the ISE (Integrated SuperBrownian Excursion).