| In this paper, we study the exclusion process and random tree by the methods of dualty, martingale, moment generating function, etc.Firstly, we study the upper bound of the occupation time variance by some knowledge of random walk and Fourier transform; Secondly, we prove the existence of exclusion process on random graph, we transform the problem into the solution of the martingale problem corresponding to the generator of the exclusion, so the problem of existenc is settled; Thirdly, we study the problem of number of vertices in nodes deletion tree process by the means of moment generating function; Finally, by the means of moment generating function and martingale methods, we study some properties of genralized random tree, including degree distribution, branching structure, degree of given vertices, maximum degree,etc.The paper contains seven chapters: In the first chapter, we mainly introduce the history of the topic we will discuss and the work we have done; In the second chapter, we mainly study an invariant formula and the equivallence of two kinds of norms induced by generator of exclusion process and its pregenerator; In the third chapter, we give an upper bound of occupation time variance by use of the invariant formula in chapter 2; In the fourth chapter, we prove the existence and uniqueness of the martingale problem corresponding to the generator of exclusion on graph; In the fifth chapter, we discuss the problem of the number of vertices of subtree when we delete a node randomly; In the sixth chapter, we generalize the random uniform recursive tree and preferential attachment tree and study the degree distribution, branching structure and maximun degree, etc. In the seventh chapter,we continue the study of random tree. The tree is formed by preferential attechment at odd time and uniform recusive attachment at even time or preferential attchment at the time of multiple of number m. We discuss the degree distribution of that tree progress.
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