The paper is motivated by \textit{S. Gogyan}'s weak thresholding version of the thresholding greedy algorithm in \(L^1 (0,1)\) with regard to the Haar system [J. Approx. Theory 161, No. 1, 49--64 (2009; Zbl 1177.41032)]. The authors extract the basic features of Gogyan's weak thresholding greedy algorithm into a new definition of branch greedy algorithms with respect to Markushevich bases in general Banach spaces. The relationship between convergence of the algorithm and uniform boundedness of the approximants is studied: these two properties are not the same in general, but are equivalent for a natural subclass of branch greedy algorithms. It is shown that, if there is a branch greedy algorithm for the system which gives the best \(n\)-term approximation up to a multiplicative constant, then the system is already greedy. Similar results are proved for branch almost greedy systems in arbitrary spaces and for branch semi-greedy Schauder bases in spaces of finite cotype.