Before revision:  We show that the fibered barcode , the fibered merge tree,_0(\mathcal{K}) is interval decomposable, then the barcode of \mathrm{H}_0(\mathcal{K}) is equal to the elder-rule-staircode.
After revision:  We show that if \mathrm{H_0(\mathcal{K}) is interval decomposable, then the barcode of \mathrm{H}_0(\mathcal{K}) is equal to the elder-rule-staircode.