Summary: Given a connected graph \(G=(V, E)\), we seek to partition the vertex set \(V\) into \(k\) non-empty parts such that the subgraph induced by each part is connected, and the partition is maximally balanced in the way that the maximum cardinality of these \(k\) parts is minimized. We refer this problem to as min-max balanced connected graph partition into \(k\) parts and denote it as \(k\)-BGP. The vertex-weighted version of this problem on trees has been studied since about four decades ago, which admits a linear time exact algorithm. The vertex-weighted \(2\)-BGP and \(3\)-BGP admit a \(5/4\)-approximation and a \(3/2\)-approximation, respectively. When \(k\ge 4\), no approximability result exists for \(k\)-BGP, i.e., the vertex unweighted variant, except a trivial \(k\)-approximation. In this paper, we present another \(3/2\)-approximation for the \(3\)-BGP and then extend it to become a \(k/2\)-approximation for \(k\)-BGP, for any fixed \(k\ge 3\). Furthermore, for \(4\)-BGP, we propose an improved \(24/13\)-approximation. To these purposes, we have designed several local improvement operations, which could find more applications in related graph partition problems.