Corollary 1 shows that reconciliation is not the right tool when a subtree of the gene tree adheres to the isolocalization property, yet there must be some information in the gene tree and species tree relationship. For instance, we expect subtrees corresponding to isorthologs in a well-supported gene tree to agree with the species tree. The following hypothesis formalizes this concept, where a tree *restricted* to a subset of leaves refers to the tree with all other leaves removed, and the edges around newly created single-child leaves contracted. In the rest of this paper, we will assume Hypothesis 2 in addition to Hypothesis 1.

### Hypothesis 2

*The gene tree G satisfies the isolocalization property and reflects the true phylogeny for the isorthogroups. Formally, for any isorthologous subset of genes* {*a*
_{
i
}
*, a*
_{
j
}
*, a*
_{
k
}}*, the tree G restricted to leaves a*
_{
i
}
*, a*
_{
j
}
*, and a*
_{
k
}
*, but relabeled by s*(*a*
_{
i
}) *= i, s*(*a*
_{
j
}) *= j, and s*(*a*
_{
k
}) *= k, agrees with the species tree restricted to genomes i, j, and k*.

We now elaborate the connection between isorthologous genes and the LCA mapping. In what follows, nodes of *G* are labeled as duplication or speciation nodes according to the LCA mapping. The elementary proof of the following lemma is omitted due to space limitations.

**Lemma 1**
*Take a pair* (*a*
_{
i
}
*, a*
_{
j
}) *of isorthologous genes, where a*
_{
i
}
*is a gene in genome i, and a*
_{
j
}
*is a gene in genome j. Then the node lca*
_{
G
}(*a*
_{
i
}, *a*
_{
j
}) *is a speciation node*.

Define a *speciation subtree* of *G* as a subtree such that all internal nodes (if any) are labeled as speciations by the LCA mapping. A corollary of Lemma 1 is that (under Hypothesis 2) isorthogroups of Γ are defined by speciation subtrees of *G*.

**Corollary 3**
*Any isorthogroup appears in G as the leaf-set of a speciation subtree*.

An *isorthologous subtree* of *G* is a speciation subtree of *G* corresponding to the leaves in an isorthogroup. Based on Corollary 3, the following definition introduces a natural alternative to reconciliation.

**Definition 6 (Isorthology Respecting History (IRH))**
*Given a gene tree G and a species tree S, a history H is an* isorthology respecting history *for* (*G*, *S*) *if and only if each isorthogroup induced by H is the leaf-set of a speciation subtree of G*.

In Figure 1, the histories *H* and *R* are isorthology respecting histories for the pair (*G*, *S*). Neither *H* nor *R* are isorthology respecting histories for the pair (*P*, *S*) since both histories imply isorthology between *a*
_{1} and *a*
_{2}, but the pair does not.

Notice that Corollary 3 does not *a priori* give us the isorthogroups for pair (*G*, *S*), as the true isorthologous subtree could be part of a larger subtree of speciation nodes. For example, the left subtree of *G* in Figure 1 is consistent with three possible configurations of isorthogroups: {{*a*
_{1}}, {*a*
_{2}}, {*a*
_{3}}},{{*a*
_{1}}, {*a*
_{2}, *a*
_{3}}}, or {{*a*
_{1}, *a*
_{2}, *a*
_{3}}}. We will call an *isorthology respecting partition* of *G* a partition
of
such that each element of
is the leaf set of a speciation subtree of *G*.

### Optimization problems

Following Corollary 3, an isorthologous respecting history appears as the most natural alternative to reconciliation. As many IRHs are possible for a given pair (*G*, *S*), an appropriate way for choosing most likely histories is required. Using a parsimony approach and either the duplication, lost, or mutation cost, the corresponding optimization problem is the following.

MINIMUM ISORTHOLOGY RESPECTING HISTORY

RECONSTRUCTION (MIRH):

**Input:** A gene tree *G* and species tree *S*.

**Output:** A *Minimum Isorthology Respecting History* (MIRH for short) for (*G*, *S*), *i.e*. an Isorthology Respecting History for (*G*, *S*) with minimum cost.

A restricted version of the MIRH problem would consider the maximal speciation subtrees of *G* as defining the isorthogroups. We later show that this isorthology respecting partition of *G* is the one that would minimize the duplication cost, but not necessarily the mutation cost.

The MIRH problem, as stated, ignores all the information on duplication and speciation nodes of *G* that are above the considered speciation subtrees. In other words, nothing is trusted in the gene tree except the isorthology information. An alternative would be to account for the hierarchy of the higher nodes in *G*.

**Definition 7 (Triplet Respecting History (TRH))**
*Let H be an isorthology respecting history for* (*G*, *S*), *and*
*be the isorthology respecting partition of G induced by H. Then H is a* triplet respecting history *if and only if for any triplet of genes* {*a, b, c*}*, where each gene is taken from a different isorthogroup of*
, *the tree G restricted to leaves a, b, and c agrees with H restricted to leaves a, b, and c*.

We can now formulate our second optimization problem.

MINIMUM TRIPLET RESPECTING HISTORY

RECONSTRUCTION (MTRH):

**Input:** A gene tree *G* and species tree *S*.

**Output:** A *Minimum Triplet Respecting History* (MTRH for short) for (*G*, *S*), *i.e*. a Triplet Respecting History for (*G*, *S*) of minimum cost.

Taking our model example in Figure 1, the true history *H* is a MIRH and a MTRH for the gene tree *G*, leading to a mutation and duplication cost of one (one duplication). Recall that *H* can never be recovered with a reconciliation when *G* respects the isolocalization property. Moreover the reconciliation *R* for *G* and *S* leads to a mutation cost that is higher (one duplication and one loss) than that of *H*. In this paper, we focus on the MIRH problem, which is the subject of the next section.