In a +-contact representation of a planar graph G, each vertex is represented as an axis-aligned plus shape consisting of two intersecting line segments (or equivalently, four axis-aligned line segments that share a common endpoint), and two plus shapes touch if and only if their corresponding vertices are adjacent in G. Let the four line segments of a plus shape be its arms. In a c-balanced representation, c ≤ 1, every arm can touch at most 「c△(」) other arms, where △ is the maximum degree of G. The widely studied T- and L-contact representations are c-balanced representations, where c could be as large as 1. In contrast, the goal in a c-balanced representation is to minimize c. Let c_k, where k ∈ {2,3}, be the smallest c such that every planar k-tree has a c-balanced representation. In this paper we show that 1/4 ≤ c_2≤ 1/3(= b_2) and 1/3 ＜ c_3 ≤ 1/2(= b_3). Our result has several consequences. Firstly, planar k-trees admit 1-bend box-orthogonal drawings with boxes of size 「b_k△(」) ×「b_k△(」) , which generalizes a result of Tayu, Nomura, and Ueno. Secondly, they admit 1-bend polyline drawings with 2「b_k△(」) slopes, which is significantly smaller than the 2△ upper bound established by Keszegh, Pach, and Palvoelgyi for arbitrary planar graphs.
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