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This work focuses on the microstructure of metallic deposits formed by galvanostatic electrodeposition inside a Hele-Shaw cell without both supporting electrolyte and flow. For a low applied current density j, the deposit grows under the form of ramified branches. As shown by Fleury (Nature, 390,, 1997), these branches are composed of small metallic crystals. This microstructure is built up by a re-nucleation process induced by the dynamics of a space charge region (non-electrically neutral solution) ahead of the growth front. When increasing j the crystal size decreases whereas the nucleation frequency increases. These latter tendencies are reversed for high j when, as experimentally observed, dendrites are formed instead of ramified branches. There must be a transition between the nucleation/growth regime (ramified branches) and the pure growth regime (dendrites). This transition is examined experimentally by carefully observing the branch microstructure by SEM. For copper and silver branches, when j is lower than a critical current density j_c (concentration-dependent), the branches are composed only of non-dendritic crystals. Whereas, when j>j_c, dendritic crystals are observed and they become the main kind of crystals constituting the branches for higher j. These observations show that the morphological transition on the pattern scale, between ramified branches and dendrites, originates from a morphological transition on the scale of the crystals constituting the branches. This latter is considered theoretically by analyzing the shape stability of the growing crystals. The Mullins & Sekerka model (shape stability of a spherical particle growing by diffusion) disagrees with these observations by predicting that the crystals are always unstable. It is proposed that the space charge layer, surrounding the growing crystals, induces a stabilizing effect.