Let \(K\) be an algebraically closed field. By an algebra we mean a basic finite dimensional \(K\)-algebra. Two algebras \(A\) and \(B\) are said to be derived equivalent if there is a triangulated category equivalence of the derived categories \(D^b(\text{mod }A)\) and \(D^b(\text{mod }B)\), where \(\text{mod }A\) and \(\text{mod }B\) are the categories of finite dimensional right \(A\)-modules and \(B\)-modules, respectively. Given a \(K\)-algebra \(A\) we denote by \(K_0(A)\) the Grothendieck group of \(\text{mod }A\). It is well-known that attaching to the isomorphism class \([X]\) of \(X\) in \(\text{mod }A\) the dimension vector \(x=\underline\dim X\) of \(X\) defines a group isomorphism \(K_0(A)\cong\mathbb{Z}^n\), where \(n\) is the number of isoclasses of simple \(A\)-modules. If \(A\) is of finite global dimension one defines the Euler quadratic form \(\chi_A\colon\mathbb{Z}^n\to\mathbb{Z}\) by the formula \(\chi(x)=\sum_{j=0}^\infty(-1)^j\dim_K\text{Ext}_A^j(X,X)\), where \(x=\underline\dim X\). It is known that if \(\chi_A\) is non-negative that under a group isomorphism of \(\mathbb{Z}^n\) the quadratic form \(\chi_A\) corresponds to a quadratic form \(q_\Delta\colon\mathbb{Z}^n\to\mathbb{Z}\) associated with a Dynkin graph \(\Delta\), which is uniquely determined by \(\chi_A\).
The main result of the paper is the following theorem. Let \(A\) be a finite dimensional \(K\)-algebra which is a tree algebra or a strongly simply connected poset algebra. Then \(A\) is derived equivalent to a tubular algebra or to an incidence \(K\)-algebra of the poset \(P(n)\) described in the paper. Moreover, if the number of isoclasses of simple \(A\)-modules is greater or equal to \(6\), then \(A\) is derived equivalent to a tubular algebra if and only if the Dynkin graph \(\text{Dyn}(\chi_A)\) associated with \(\chi_A\) is of type \(\mathbb{E}_p\) with \(p=6,7,8\), and \(A\) is derived equivalent to the incidence \(K\)-algebra \(KP(n)\) of the poset \(P(n)\) if and only if the Dynkin graph \(\text{Dyn}(\chi_A)\) is of type \(\mathbb{D}_{n-2}\).