A scaling exponent of 0.472 was obtained from some experimental observations of bi-dimensional seaweeds for their length-biomass allometry. It was sufficiently higher than the value of 0.25, which was almost regarded as the universal exponent, on the basis of the experimental data obtained for unicellular microalgae and vascular plants. Later, a simple theoretical model predicted a length-biomass exponent of 0.5, which was consistent, to a certain extent, with the data for bi-dimensional seaweeds. The present article has a theoretical study based on an assumption of a power-law relation between the two length parameters of a bi-dimensional organism, upon which the scaling exponent has been found to depend. The length-biomass scaling exponents can be expressed in terms of this power index. Mathematical expressions of biomass, of a bi-dimensional organism, in terms of its two length parameters, have been formulated here, which would be useful for experimental studies on allometric scaling. A theoretical model has been proposed to explain the origin of the scaling exponent of 0.25, observed for vascular plants. An attempt has been made to extend this model for two dimensional species, to three dimensional species, by assuming the existence of allometric relations among the length parameters along three mutually perpendicular directions.