In ‘Groups acting on connected cubes and Kazhdan’s property T’, [29], Niblo and Roller showed that any CAT(0) cube complex embeds combinatorially and quasi-isometrically in the Hilbert space `2(H) where H is the set of hyperplanes. This Hilbert space may be viewed as the completion of an infinite product of trees. In this thesis, we consider the question of the existence of quasi-isometric maps from CAT(0) cube complexes to finite products of trees, restricting our attention to folding maps as used in [29]. Following an overview of the properties of CAT(0) cube complexes, we first prove that there exists CAT(0) square complexes which do not fold into a product of trees with fewer than k factors for arbitrary k, giving examples which admit co-compact proper actions by right-angled Coxeter groups. We also show that there exists a CAT(0) square complex which does not fold into any finite product of trees. We then identify a class of group actions on CAT(0) cube complexes for which the existence of such an action implies the existence of a quasi-isometric embedding of that group in a finite product of finitely branching trees. We apply this result to surface groups, certain 3-manifold groups and more generally to Coxeter groups which do not contain affine triangle subgroups.

Spanning tree congestion was defined by Ostrovskii (2004) as a measure of how well a network can perform if only minimal connection can be maintained. We compute the parameter for several families of graphs. In particular, by partitioning a hypercube into pieces with almost optimal edge-boundaries, we give tight estimates of the parameter thereby disproving a conjecture of Hruska (2008). For a typical random graph, the parameter exhibits a zigzag behaviour reflecting the feature that it is not monotone in the number of edges. This motivates the study of the most congested graphs where we show that any graph is close to a graph with small congestion. Next, we enumerate independent sets. Using the independent set polynomial, we compute the extrema of averages in trees and graphs. Furthermore, we consider inverse problems among trees and resolve a conjecture of Wagner (2009). A result in a more general setting is also proved which answers a question of Alon, Haber and Krivelevich (2011). After briefly considering polynomial invariants of general graphs, we specialize into trees. Three levels of tree distinguishing power are exhibited. We show that polynomials which do not distinguish rooted trees define typically exponentially large equivalence classes. On the other hand, we prove that the rooted Ising polynomial distinguishes rooted trees and that the Negami polynomial determines the subtree polynomial, strengthening results of Bollobás and Riordan (2000) and Martin, Morin and Wagner (2008). The top level consists of the chromatic symmetric function and it is proved to be a complete invariant for caterpillars.

Cette thèse comporte deux parties indépendantes. Dans la première partie, nous nous intéressons à l’analyse asymptotique de quelques statistiques sur les arbres binaires de recherche (ABR). Dans la deuxième partie, nous nous intéressons à l’étude du problème du collectionneur de coupons impatient. Dans la première partie, en suivant le modèle introduit par Aguech, Lasmar et Mahmoud [Probab. Engrg. Inform. Sci. 21 (2007) 133—141], on définit la profondeur pondérée d’un nœud dans un arbre binaire enraciné étiqueté comme la somme de toutes les clés sur le chemin qui relie ce nœud à la racine. Nous analysons alors dans ABR, les profondeurs pondérées des nœuds avec des clés données, le dernier nœud inséré, les nœuds ordonnés selon le processus de recherche en profondeur, la profondeur pondérée des trajets, l’indice de Wiener pondéré et les profondeurs pondérées des nœuds avec au plus un enfant. Dans la deuxième partie, nous étudions la forme asymptotique de la courbe de la complétion de la collection conditionnée à T_n≤ (1+Λ), Λ>0, où T_n≃n ln⁡n désigne le temps nécessaire pour compléter la collection. Puis, en tant qu’application, nous étudions les automates déterministes et accessibles et nous fournissons une nouvelle dérivation d’une formule due à Korsunov [Kor78, Kor86] / This Phd thesis is divided into two independent parts. In the first part, we provide an asymptotic analysis of some statistics on the binary search tree. In the second part, we study the coupon collector problem with a constraint. In the first part, following the model introduced by Aguech, Lasmar and Mahmoud [Probab. Engrg. Inform. Sci. 21 (2007) 133—141], the weighted depth of a node in a labelled rooted tree is the sum of all labels on the path connecting the node to the root. We analyze the following statistics : the weighted depths of nodes with given labels, the last inserted node, nodes ordered as visited by the depth first search procees, the weighted path length, the weighted Wiener index and the weighted depths of nodes with at most one child in a random binary search tree. In the second part, we study the asymptotic shape of the completion curve of the collection conditioned to T_n≤ (1+Λ), Λ>0, where T_n≃n ln⁡n is the time needed to complete accessible automata, we provide a new derivation of a formula due to Korsunov [Kor78, Kor86]