we study the limiting distribution of the degree of a given node in a scaled attachment random recursive tree, a generalized random recursive tree, which is introduced by devroye et. al (2011). in a scaled attachment random recursive tree, every node $i$ is attached to the node labeled $lfloor ix_i floor$ where $x_0$, $ldots$ , $x_n$ is a sequence of i.i.d. random variables, with support in [0,...

In this paper we study the first Zagreb index in bucket recursive trees containing buckets with variable capacities. This model was introduced by Kazemi in 2012. We obtain the mean and variance of the first Zagreb index and introduce a martingale based on this quantity.

in this paper, using generalized {polya} urn models we find the expected value of the size of a branch in recursive $k$-ary trees. we also find the expectation of the number of nodes of a given outdegree in a branch of such trees.

‎The first variable Zagreb index of graph $G$ is defined as‎ ‎begin{eqnarray*}‎ ‎M_{1,lambda}(G)=sum_{vin V(G)}d(v)^{2lambda}‎, ‎end{eqnarray*}‎ ‎where $lambda$ is a real number and $d(v)$ is the degree of‎ ‎vertex $v$‎. ‎In this paper‎, ‎some upper and lower bounds for the distribution function and expected value of this index in random increasing trees (rec...

Abstract In this work we analyse bucket increasing tree families. We introduce two simple stochastic growth processes, generating random trees of size n , complementing the earlier result Mahmoud and Smythe (1995, Theoret. Comput. Sci. 144 221–249.) for recursive trees. On combinatorial side, define multilabelled generalisations families d -ary generalised plane-oriented Additionally, a cluster...

We study the limiting distribution of the degree of a given node in a scaled attachment random recursive tree, a generalized random recursive tree, which is introduced by Devroye et. al (2011). In a scaled attachment random recursive tree, every node $i$ is attached to the node labeled $lfloor iX_i floor$ where $X_0$, $ldots$ , $X_n$ is a sequence of i.i.d. random variables, with support in [0,...

consider the random walk among n places with n(n - 1)/2 transports. we attach an exponential random variable xij to each transport between places pi and pj and take these random variables mutually independent. if transports are possible or impossible independently with probability p and 1-p, respectively, then we give a lower bound for the distribution function of the smallest path at point log...

Bucket increasing trees are multilabelled generalizations of trees, where each non-leaf node carries b labels, with b≥1 a fixed integer. We provide fundamental result, giving complete characterization all families bucket that can be generated by tree evolution process. also several equivalent properties, complementing and extending earlier results for ordinary to trees. Additionally, we state s...

In this paper, using generalized {polya} urn models we find the expected value of the size of a branch in recursive $k$-ary trees. We also find the expectation of the number of nodes of a given outdegree in a branch of such trees.