The inverse degree index of a simple connected graph \( G \) is defined by \[ \mathrm{ID}(G)=\sum_{v\in V(G)} \frac{1}{\deg_G(v)}. \] This index, also known as the modified first Zagreb index or the Randi\'c index of order \( -1 \), has attracted considerable attention in chemical graph theory and discrete mathematics. Recent studies on degree--based indices, including various Zagreb and Mostar variants [3, 4, 13], further highlight the relevance of extremal problems involving vertex degrees. It is known that among all connected graphs of order \( n \) the star \( K_{1,n-1} \) uniquely maximizes the inverse degree index [12]. We investigate the analogous extremal problem under a maximum degree constraint: given integers \( n\ge 3 \) and \( 2\le \Delta\le n-1 \), which tree on \( n \) vertices with maximum degree at most \( \Delta \) maximizes the inverse degree? We show that the unique extremal tree is always a double star. Concretely, writing \( \mathrm{DS}(a,b) \) for the tree obtained by joining two vertices \( u \) and \( v \) with an edge and attaching \( a-1 \) leaves to \( u \) and \( b-1 \) leaves to \( v \), we prove \[ \mathrm{ID}(T)\;\le\;(n-2)+\frac{1}{\Delta}+\frac{1}{n-\Delta} \] for every tree \( T \) on \( n \) vertices with maximum degree at most \( \Delta \), with equality if and only if \( T\cong \mathrm{DS}(\Delta,n-\Delta) \) or \( T\cong \mathrm{DS}(n-\Delta,\Delta) \). Our argument uses a simple degree-transfer operation to monotonically increase the inverse degree index without violating the degree constraint. We also present computational experiments for \( n\le 8 \) that support the theorem and illustrate the extremality of double stars. Several open questions on inverse degree indices of general graphs are discussed.