In this thesis we look at various tools to analyse eigenvalues and eigenvectors and use themto prove the following main results. 1. We show that given any matrix A, there is a small perturbation of the matrix such that post perturbation, the matrix is almost normal. In particular there exists E, with ||E|| ≤ δ||A|| such that A + E is diagonalizable and its eigenvector matrix has polynomially (in 1/δ and n) bounded condition number. 2. We prove a necessary and sufficient condition for any local periodic operator on the universal cover of a finite graph to have a point spectrum. In particular we show that for λ to be in the point spectrum, the base graph must admit an induced forest with a very specific combinatorial structure and that the induced operator on it must also have λ as an eigenvalue. To prove the first result we study the volume of the pseudospectrum with the help of some tools from stochastic calculus. Along the way we also see why it implies a conjecture by Sankar, Spielman and Teng on the optimal constant for smoothed analysis of condition numbers. For the second result we show that a condition conjectured by Aomoto to be necessary and sufficient for the existence of point spectrum of certain operators on periodic trees is indeed so. Aomoto had already shown why the condition was necessary. We give a more intuitive proof of it and along the way also show sufficiency.

We investigate scaling limits of several types of random trees. The study of scaling limits of random trees was initiated by Aldous in the early 1990's. One of the primary goals of his work was to study the uniform distribution on trees with n vertices as n grew to infinity, as well as uniform distributions on other combinatorially motivated models of trees with n vertices. We are motivated by studying combinatorial models of random trees with n leaves as n goes to infinity. Conditioning on the number of leaves rather than the number of vertices has significant consequences in terms of what techniques are applicable. We deal with these issues by developing a general theory for scaling limits of Markov branching trees whose size is given by their number of vertices with out-degree in a fixed set. This general theory is then applied to obtain scaling limits of Galton-Watson trees conditioned on their number of vertices with out-degree in a fixed set. We also show that many combinatorial models of trees with n leaves can be realized as Galton-Watson trees (or probabilistic transforms thereof) conditioned to have n leaves.

Exchangeability -- the probabilistic symmetry meaning ``invariance under the action of the symmetric group,'' or less formally, ``irrelevance of labels or indices'' -- has been the subject of continuing interest to probabilists and statisticians since de Finetti's celebrated characterization of infinite exchangeable sequences of Bernoulli random variables as mixtures of IID sequences. The topic of this dissertation is exchangeability as it pertains to random partitions and trees.

The main result is a de Finetti-type theorem characterizing a class of exchangeable trees called hierarchies <\italic> which arise in connection with fragmentation processes and hierarchical clustering problems. The other results are somewhat related in that they involve consideration of moments of statistics of exchangeable partitions or trees. One of these concerns random trees with leaves labeled by consecutive natural numbers which are exchangeable in the sense that deterministic permutation of the leaf labels does not change the distribution of the tree. In such trees, the set of interleaf distances is exchangeable, and so, for example, the distance between leaf 1 and leaf 2 is equal in distribution to the distance between leaf 2 and leaf 3. Distributional constraints of this type arising from exchangeability can be used to characterize finite dimensional marginals of well-understood trees such as the Brownian CRT. As an application we show that the Brownian CRT is the scaling limit of uniform random hierarchies. Another result is the characterization of the two-parameter family of Ewens-Pitman partitions by a kind of deletion property: speaking loosely, the Ewens-Pitman family is the class of exchangeable partitions in which the block containing 1 carries no information about the rest of the partition.