  We derive a set of genuine multi-mode entanglement criteria for second
moments of the quadrature operators. The criteria have a common form of the
uncertainty relation between sums of variances of position and momentum
quadrature combinations. A unique feature of the criteria is that the sums
contain the least possible number of variances of at most two-mode
combinations. The number of second moments we need to know to apply the
criteria thus scales only linearly with the number of modes, as opposed to the
quadratic scaling of the already existing criteria. Each criterion is
associated with a tree graph, which allowed us to develop a direct method of
construction of the criteria based solely on the structure of the underlying
tree. The practicality of the proposed criteria is demonstrated by finding a
number of examples of Gaussian states of up to six modes, whose genuine
multi-mode entanglement is detected by them. The designed criteria are
particularly suitable for verification of genuine multipartite entanglement in
large multi-mode states or when only a set of two-mode nearest-neighbour
marginal covariance matrices of the investigated state is available.
