The canonical Wnt/β-catenin pathway in Hydra governs the head organizer at the hypostome, acting as a key morphogenetic signal for regeneration and axis formation. Experimental overexpression of β-catenin downstream effectors allows researchers to study the effects on body patterning and morphogenesis.

Consequences of Global β-Catenin Overexpression

• Ectopic Bud Formation:
  Over-activation or global mis-expression of β-catenin leads to multiple and misplaced organizer sites along the Hydra body column.

• Multiple Axes Development:
  These ectopic organizers induce buds and secondary axes at various points, including on top of new buds, causing a branched morphology.

• Excess Head-Like Structures:
  The elevated Wnt/β-catenin activity mimics head organizer function, resulting in repetitive head and bud formations.

• Experimental Corroboration:
  Studies show transgenic Hydra lines with β-catenin overexpression develop branches or ectopic heads spaced naturally but distributed more abundantly than wild-type animals.

Explanation of Other Options

• Simple ectopic tentacles formation (option 2) is part of the phenotype but incomplete as the main effect includes ectopic buds and axes.

• Both ectopic tentacles and buds (option 3) is close but option 1 more fully captures the phenomenon of ectopic buds forming along the entire axis and on newly formed buds themselves.

• No change (option 4) is contradicted by extensive experimental evidence.

Summary

In transgenic Hydra with mis-expressed β-catenin, the organism shows ectopic buds all along the body axis and even atop newly formed buds, reflecting the powerful role of β-catenin in establishing organizers and driving regenerative growth.

Final Answer:
(1) Ectopic buds will be formed all along the body axis and even on the top of the newly formed buds.