The definable tree property for successors of cardinals

Strengthening a result of Amir Leshem [7], we prove that the consistency strength of holding GCH together with definable tree property for all successors of regular cardinals is precisely equal to the consistency strength of existence of proper class many Π 1 reflecting cardinals. Moreover it is proved that if κ is a supercompact cardinal and λ > κ is measurable, then there is a generic extension of the universe in which κ is a strong limit singular cardinal of cofinality ω, λ = κ, and the definable tree property holds at κ. Additionally we can have 2κ > κ, so that SCH fails at κ.