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We introduce the concept of Random Sequential Renormalization (RSR) forarbitrary networks. RSR is a graph renormalization procedure that locallyaggregates nodes to produce a coarse grained network. It is analogous to the(quasi-)parallel renormalization schemes introduced by C. Song {\it et al.}(Nature {\bf 433}, 392 (2005)) and studied more recently by F. Radicchi {\it etal.} (Phys. Rev. Lett. {\bf 101}, 148701 (2008)), but much simpler and easierto implement. In this first paper we apply RSR to critical trees and deriveanalytical results consistent with numerical simulations. Critical treesexhibit three regimes in their evolution under RSR: (i) An initial regime$N_0^{\nu}\lesssim N<N_0$, where $N$ is the number of nodes at some step in therenormalization and $N_0$ is the initial size. RSR in this regime is describedby a mean field theory and fluctuations from one realization to another aresmall. The exponent $\nu=1/2$ is derived using random walk arguments. Thedegree distribution becomes broader under successive renormalization --reaching a power law, $p_k\sim 1/k^{\gamma}$ with $\gamma=2$ and a variancethat diverges as $N_0^{1/2}$ at the end of this regime. Both of these resultsare derived based on a scaling theory. (ii) An intermediate regime for$N_0^{1/4}\lesssim N \lesssim N_0^{1/2}$, in which hubs develop, andfluctuations between different realizations of the RSR are large. Crossoverfunctions exhibiting finite size scaling, in the critical region $N\simN_0^{1/2} \to \infty$, connect the behaviors in the first two regimes. (iii)The last regime, for $1 \ll N\lesssim N_0^{1/4}$, is characterized by theappearance of star configurations with a central hub surrounded by many leaves.The distribution of sizes where stars first form is found numerically to be apower law up to a cutoff that scales as $N_0^{\nu_{star}}$ with$\nu_{star}\approx 1/4$.

Random Sequential Renormalization of Networks I: Application to Critical Trees