on the first variable zagreb index

‎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 (recursive trees‎, ‎plane-oriented recursive trees and binary increasing trees) are‎ ‎given‎.