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We study the problems of leader election and population size counting forpopulation protocols: networks of finite-state anonymous agents that interactrandomly under a uniform random scheduler. We show a protocol for leaderelection that terminates in $O(\log_m(n) \cdot \log_2 n)$ parallel time, where$m$ is a parameter, using $O(\max\{m,\log n\})$ states. By adjusting theparameter $m$ between a constant and $n$, we obtain a single leader electionprotocol whose time and space can be smoothly traded off between $O(\log^2 n)$to $O(\log n)$ time and $O(\log n)$ to $O(n)$ states. Finally, we give aprotocol which provides an upper bound $\hat{n}$ of the size $n$ of thepopulation, where $\hat{n}$ is at most $n^a$ for some $a>1$. This protocolassumes the existence of a unique leader in the population and stabilizes in$\Theta{(\log{n})}$ parallel time, using constant number of states in everynode, except the unique leader which is required to use $\Theta{(\log^2{n})}$states.

Fast Approximate Counting and Leader Election in Populations