The streamers of glowing, ionized gas in a lightning strike have an inevitable tendency to break up at their tips, creating a familiar tree-like pattern. This tip-splitting is similar to the same kind of process that gives rise to branching in snowflakes, river channels, tumor growth and other different natural phenomena. In the 1980s, researchers identified a process called Laplacian growth which explains why branching occurs at the boundary between two substances. We have shown that a Laplacian instability also occurs at the edge of a lightning streamer. We analyse the minimal continuum model for a negative streamer in a non-ionized and non-attaching gas with impact ionization reaction in local field approximation. Using computer simulations, we have found that a streamer undergoes spontaneous branching at the tip. The mathematical analysis of the model identifies the instabilities as Laplacian ones. Despite their complicated structure, the basic model for streamers is simple. A single free electron traveling in a strong, uniform electric field ionizes the gaseous molecules around it, generating more electrons and a chain reaction of ionization. The ionized gas creates its own electric field, and a streamer is born. Eventually its charge and field concentrate in a thin, dome-shaped layer at the tip of the growing streamer. In this state, any tiny protrusions in the charged front focus the electric field around them. Because the speed of ionization depends on the surrounding field, these bumps grow faster than the rest of the front, and branching results. A similar mechanism can operate in a coral reef, when one part pulls ahead of the rest, it gets more food and so grows even faster.